This case returns to us from the district court following (1) a judgment of this Court ruling that portions of Article Y, § 6, of the New York State Constitution and § 85 of the New York Civil Service Law were unconstitutional, (2) the affirmance of that judgment by the United States Supreme Court, see Soto-Lopez v. New York City Civil Service Commission, 755 F.2d 266 (2d Cir. 1985) (“Soto-Lopez /”), aff'd sub nom. Attorney General v. Soto-Lopez, 476 U.S. 898, 106 S.Ct. 2317, 90 L.Ed. 2d 899 (1986), and (3) a January 5, 1987 judgment of the United States District Court for the Southern District of New York (“1987 Judgment”), Richard Owen, Judge, purporting to reflect the final adjudication of this action. In accordance with the Supreme Court’s affirmance of Soto-Lopez I, the 1987 Judgment declared unconstitutional the portions of the above provisions of New York law that denied to veterans who were not residing in New York State when they entered the armed services the civil service eligibility bonus points that were given to veterans who *165were so residing; the 1987 Judgment ordered that plaintiffs Eduardo Soto-Lopez and Eliezer Baez-Hernandez be granted veterans’ preference points if they are qualified to receive them and be given appointments with retroactive seniority if they meet other requirements for appointment.

Plaintiffs appeal from so much of the 1987 Judgment as (1) denied permanent injunctive relief, contending principally that that denial did not comply with the mandate of this Court in Soto-Lopez I, and (2) denied them backpay. Defendants New York City Civil Service Commission, et al. (collectively the “City defendants”), urge us either (a) to affirm the denial of injunc-tive relief principally on the ground that such relief is unnecessary because they have been complying with the mandate of Soto-Lopez I since March 1985, approximately one month after that decision was rendered, or (b) to rule that any permanent injunction should not bar their use of current eligibility lists compiled prior to March 1985 in reliance on the unconstitutional provisions. They urge us to affirm the denial of backpay both on the ground of immunity because, in denying veterans’ preference points to plaintiffs, the City defendants had merely followed the then-presumptively valid provisions of state law, and on the ground that plaintiffs have not demonstrated their individual eligibility for the ppsi-tions to which they sought appointment.

On its face, the 1987 Judgment does not comply with the mandate of this Court in Soto-Lopez I. Accordingly, we vacate the judgment in part and remand for further proceedings leading to the entry of permanent injunctive relief against the City defendants and to the final adjudication of plaintiffs’ claims.

I. BACKGROUND

A. The Invalidation of the Past-Residency Requirements

The background of the present litigation is set forth in detail in Soto-Lopez I, 755 F.2d 266, and will be but briefly summarized here. The state-law provisions at issue granted bonus points to otherwise qualified civil service position applicants who were veterans of the United States armed services, but only if they were residents of New York when they were inducted into the armed services (“past-residency requirements”). N.Y. Const, art. V, § 6; N.Y.Civ.Serv.Law § 85 (McKinney 1983). Plaintiffs, veterans who had applied for positions covered by civil service, were denied veterans’ bonus points on their civil service examinations because they had not been residing in New York when they entered the armed services. They commenced the present action (1) to have the past-residency requirements declared unconstitutional, (2) to enjoin the enforcement of those provisions against otherwise qualified veterans, and (3) to compel the City defendants to award plaintiffs veterans’ preference points, adjust their standing on job eligibility lists, and appoint them to positions for which they had applied, with seniority and backpay.

The district court dismissed the complaint on the ground that the residency-at-time-of-induction distinction had been upheld against constitutional challenge in August v. Bronstein, 369 F.Supp. 190 (S.D.N.Y.) (three-judge court), aff'd mem., 417 U.S. 901, 94 S.Ct. 2596, 41 L.Ed.2d 208 (1974). In Soto-Lopez I, this Court reversed, ruling that August v. Bronstein had been implicitly overruled by Zobel v. Williams, 457 U.S. 55, 102 S.Ct. 2309, 72 L.Ed.2d 672 (1982), and that the residency distinction violated plaintiffs’ constitutional right to equal protection and right to travel. We remanded to the district court (1) for entry of a judgment permanently enjoining defendants from denying veterans’ bonus points to “otherwise qualifying veterans solely on the ground that they were not New York residents at the time of their induction into the United States armed forces,” 755 F.2d at 282, and (2) for further proceedings to determine any issues relating to plaintiffs’ requests for individual relief, including “whether either plaintiff should receive immediate appointment, if so to what position, and whether awards of backpay or retroactive seniority are appropriate, as well as issues relating to defend*166ants’ defenses of immunity,” id. at 281; see id. at 281-82.

The State of New York (“State”), which had intervened in the action to defend the constitutionality of the challenged constitutional and statutory provisions, appealed Soto-Lopez I to the Supreme Court. In July 1985, the district court entered an order that, inter alia, “permanently enjoined” the City defendants from enforcing the residency-at-time-of-induction requirements “pending the outcome of the [State’s] appeal.” In June 1986, the Supreme Court affirmed our judgment that the past-residency requirements were unconstitutional. 106 S.Ct. 2317.

B. The 1987 Judgment

Following the decision of the Supreme Court, the district court entered the 1987 Judgment, which contains no injunction against enforcement of the past-residency requirements. That Judgment, after declaring the pertinent New York provisions unconstitutional “in accordance with the opinion of the United States Supreme Court,” provides as follows:

1. Defendants New York City Civil Service Commission, New York City Department of Personnel, Mark Lebow, and Juan Ortiz (“City, defendants” [sic]) shall award plaintiffs veterans credits pursuant to section 85 of the Civil Service Law if plaintiffs meet the requirements of section 85 Law [sic ] other than the requirement that they were “a resident of [the State of New York] at the time of entrance into the armed forces;

2. If plaintiff Soto-Lopez invokes his entitlement to veterans credits on the eligible list from examination # 5003 for the position of housing caretaker or any other eligible list on an examination taken after the filing of the Complaint on February 22,1983, and if with these additional credits plaintiff Soto-Lopez would have been appointed or promoted on these eligibility lists, City Defendants shall appoint or promote plaintiff Soto-Lopez with back seniority as of the date that he would have been appointed or promoted with veterans credits, provided he is otherwise qualified for the position;

3.If plaintiff Baez-Hernandez invokes his entitlement to veterans credits on the eligibility list from examination # 8144 for the position of O.C. Research Assistant or on the eligibility list from examination #0153 for Assistant Accountant, or on any other examination taken after the filing of the Complaint on February 22,1983, and if with these additional credits plaintiff Baez-Hernandez would have been appointed or promoted on these eligible lists, City Defendants shall appoint or promote plaintiff Baez-Hernandez with back seniority as of the date that he would have been appointed or promoted with veterans credits, provided he is otherwise qualified for the position.

1987 Judgment at 2-3 (bracketed phrase and lack of end quote in original). The remainder of the 1987 Judgment provides that plaintiffs are entitled to an award of attorney’s fees and costs “[a]s prevailing parties,” in an amount to be agreed by the parties or, failing such an agreement, determined by the district court.

The record leading to the entry of judgment on these terms is sparse. Papers submitted to the district judge were not formally filed with the district court and do not appear in the record on appeal. These papers, copies of some of which were provided to this Court after oral argument of this appeal, included at least one proposed order and letter of argument from each group of parties, i.e., the plaintiffs, the City defendants, and the State. Apparently no factual presentations were made, and no evidentiary hearings were held either with respect to appropriate limitations on the scope of injunctive relief or with respect to the entitlement of the plaintiffs to individual relief.

The proposed judgments submitted by both the City defendants and the plaintiffs included a provision that would have permanently enjoined the City defendants from enforcing the past-residency requirements. In entering the 1987 Judgment, the court appears to have adopted verbatim the *167order proposed by the State (including typographical idiosyncrasies), which omitted injunctive relief. In support of the denial of permanent injunctive relief, the State argued, inter alia, (1) that injunctive relief was “unnecessary. New York State will comply with the law”; (2) that “to the extent that the proposed injunction would apply to persons other than plaintiffs, it would be improper since this is not a class action”; and (3) that both (a) an injunction against the City defendants’ enforcing the past-residency provisions and (b) a further order, as proposed by plaintiffs, that the City defendants take appropriate action to publicize the judgment and allow veterans on existing civil service lists to exercise their rights to claim veterans’ preference points if they meet other eligibility requirements, would be “hopelessly vague and could be interpreted ... to require the City defendants to take away appointments and promotions made in reliance on [the invalidated provisions before they were invalidated].” (State’s October 21, 1986 letter to Hon. Richard Owen.)

The district court gave no reasons, either in writing or orally, for denying injunctive relief and fashioning the 1987 Judgment as it did.

C. Issues on This Appeal

Following entry of the 1987 Judgment, plaintiffs appealed, contending that the court should have entered a permanent injunction and should have gi anted them backpay. They argue that their entitlement to appointments was established by their factual submissions in support of their motion for summary judgment prior to Soto-Lopez I and by the failure of the City defendants to controvert those submissions either at that time or following our remand. See Soto-Lopez I, 755 F.2d at 281 (directing that defendants be given another opportunity to controvert plaintiffs’ statements pursuant to Local Rule 3(g) “[i]f defendants believe there is any genuine issue as to whether either plaintiff met the other criteria for the award of the bonus points claimed”).

The State, whose proposed order was adopted by the district court, has not participated in this appeal. The City defendants, though their own proposed order contained a provision for injunctive relief, have now largely adopted the positions previously articulated by the State in opposition to the granting of injunctive relief. In addition, they contend that any injunctive relief must be limited because state law prohibits them from altering the present sequence of candidates on civil service eligibility lists. With respect to plaintiffs’ claims for individual relief, the City defendants contend that plaintiffs have not shown their eligibility for appointments and a fortiori are not entitled to awards of backpay. In addition, they claim that their good-faith reliance on August v. Bronstein entitles them to immunity from any claim for backpay.

For the reasons below, we conclude that the district court erred in failing to follow the mandate of this Court to award permanent injunctive relief and to finally adjudicate plaintiffs’ claims for individual relief. Accordingly, we remand for further proceedings.

II. DISCUSSION

A. The Law of the Case With Respect to Injunctive Relief

Under the law-of-the-case doctrine, “ ‘[w]here a case has been decided by an appellate court and remanded, the court to which it is remanded must proceed in accordance with the mandate and such law of the case as was established by the appellate court.’ ” United States v. Fernandez, 506 F.2d 1200, 1202 (2d Cir. 1974) (quoting IB J. Moore, Federal Practice If 0.404[10], at 571 (2d ed. 1974) (footnotes omitted)). It has long been established that this doctrine imposes a duty on a lower court to follow a ruling made by the reviewing court at an earlier stage of a case, and that the lower court has no discretion to disregard that duty. See, e.g., In re Sanford Fork & Tool Co., 160 U.S. 247, 255, 16 S.Ct. 291, 293, 40 L.Ed. 414 (1895); United States v. Fernandez, 506 F.2d at 1202-03.

*168In Soto-Lopez I, we remanded to the district court with instructions for, inter alia, the entry of judgment “permanently enjoining defendants from denying preference points to” “otherwise qualifying veterans solely on the ground that they were not New York residents at the time of their induction into the United States armed forces.” 755 F.2d at 282. Our prior decision thus imposed on the district court a duty to enter a permanent injunction against the City defendants’ enforcement of the past-residency requirements, and that court was not entitled to disregard this instruction.

The law-of-the-case doctrine does not deprive an appellate court of discretion to reconsider its own prior rulings, see, e.g., United States v. Fernandez, 506 F.2d at 1203; but prudential concerns for the finality of judgments make it sound practice to exercise that discretion only sparingly and to adhere to the prior rulings unless cogent reasons warrant a departure, id. In the present case, leaving aside for the moment the precise terms of the permanent injunction, a matter not addressed in Soto-Lopez I but discussed in Part II.B. below, no sound reason has been advanced for the elimination of the injunctive relief ordered in Soto-Lopez I.

The argument that a permanent injunction should be denied because the City defendants since March 1985 have discontinued enforcement of the unconstitutional provisions is unpersuasive. There can be no doubt that “the court’s power to grant injunctive relief survives discontinuance of the illegal conduct,” United States v. W.T. Grant Co., 345 U.S. 629, 633, 73 S.Ct. 894, 897, 97 L.Ed. 1303 (1953), and that it is appropriate to exercise that power when “there exists some cognizable danger of recurrent violation,” id. In the present case, defendants’ “discontinuance” of enforcement of the past-residency requirements has been highly selective, being limited solely to the construction of new civil service eligibility lists when new examinations are held. Where there has been no new examination, the City defendants have continued to use old eligibility lists, compiled using the unconstitutional provisions, to deny veterans’ bonus points — and hence higher rankings and undoubtedly appointments to jobs — to otherwise eligible veterans solely because they were not residents of New York when they entered the armed services. Indeed, both the City defendants and the State have argued against injunc-tive relief precisely because they wish to continue to use current lists that reflect the past-residency requirements. Hence the “danger” of defendants’ continued use of the unconstitutional provisions is, in this case, a reality.

The City defendants also argue that an injunction, because it would grant the rights we have upheld for plaintiffs to qualified veterans other than plaintiffs, would be improper because plaintiffs did not bring this suit as a class action. This argument is meritless. Although plaintiffs did not seek to have the case proceed as a class action, they requested relief of general application, asking that the court “permanently enjoin[] defendants from denying ... [bonus] points” to “otherwise qualifying veterans who were not residents of New York at the time of their entry into the armed forces,” Soto-Lopez I, 755 F.2d at 269 n. 1. The availability of such injunctive relief is not limited to class actions. When a state statute has been ruled unconstitutional, state actors have an obligation to desist from enforcing that statute. E.g., Cooper v. Aaron, 358 U.S. 1, 17-18, 78 S.Ct. 1401, 1409-10, 3 L.Ed.2d 5 (1958). Thus, when it has been held unconstitutional to deny benefits to otherwise qualified persons on the ground that they are members of a certain group, the officials have the obligation to cease denying those benefits not just to the named plaintiffs but also to all other qualified members of the group. We see no reason to alter the view implicit in Soto-Lopez I that an injunction is an appropriate remedy especially when, as here, it is conceded that the officials will otherwise continue to enforce the unlawful provisions against some who are not parties to the suit. Given an established unconstitutionality, it would be, in the words of Judge Friendly, “unthinkable” to permit *169the officials to “insist on other actions being brought.” See Vulcan Society v. Civil Service Commission, 490 F.2d 387, 399 (2d Cir. 1973). In such circumstances, injunc-tive relief is appropriate without the recognition of a formal class, for “insofar as the relief sought is prohibitory, an action seeking declaratory or injunctive relief against state officials on the ground of unconstitutionality of a statute or administrative practice is the archetype of one where class action designation is largely a formality, at least for the plaintiffs.” Galvan v. Levine, 490 F.2d 1255, 1261 (2d Cir. 1973), cert. denied, 417 U.S. 936, 94 S.Ct. 2652, 41 L.Ed.2d 240 (1974).

Nor are we persuaded by the City defendants’ argument that injunctive relief should not issue because state law provides that application for veterans’ bonus points must be made “between the date of ... application for examination and the date of the establishment of the resulting eligible list.” N.Y.Civ.Serv.Law § 85.3; see also N.Y. Const, art. V, § 6 (veterans’ preference points “shall be granted only at the time of establishment of an eligible list”). While, at first blush, this might appear to be a contention that otherwise qualified veterans who did not claim veterans’ bonus points prior to compilation of the eligibility lists have waived their rights, it is revealing that defendants also continue to refuse to recognize the rights of veterans who timely claimed these points and were denied them. (City defendants’ March 28, 1985 letter to Hon. Richard Owen.) To the extent that the City defendants seek to make a waiver argument, we think it ill becomes them, as they seek to limit their liability by invoking their own reliance on August v. Bronstein, to contend that veterans who failed to claim the bonus points may not be excused for that failure in light of August v. Bronstein. To the extent that the thrust of the City defendants’ argument is simply that state law forbids the City defendants to respect the recently judicially declared federal constitutional rights, that argument must be rejected as contrary to the Supremacy Clause of the Constitution.

Finally, we reject the argument that in-junctive relief requiring the granting of veterans’ bonus points to those previously denied such points because of the past-residency requirements would be inappropriate because it would deprive others of positions and promotions they have received. We did not suggest in Soto-Lopez I, ánd we do not suggest now, that defendants remove any person from a position already awarded. We simply remain unpersuaded that the City defendants should be allowed to continue for the indefinite future to fill positions with applicants listed in a sequence that reflects the unconstitutional denial to some candidates of the veterans’ bonus points.

B. The Terms of the Injunction

Our instruction in Soto-Lopez I did not specify the terms of the injunction to be entered, and in light of the lack of any development whatever of a factual record following that remand, it is not entirely clear what the precise terms of such an injunction should be. Certain minima, however, seem clear.

At the very least, we think the City defendants should be barred from denying preference points to any qualified veteran who, not having received such points previously because of the past-residency requirements, comes forward to request such points. In addition, the City defendants should be prohibited from continuing indefinitely to use pre-1985 eligibility lists to deny qualified veterans the preference points they could have received but for the unconstitutional past-residency requirements. In setting these minimum requirements, however, we reiterate that the injunction should not require that any person already appointed to a position be deprived of that position.

To accomplish the prohibition against the City defendants’ continued use of the unconstitutional provisions, defendants should be required to give some form of notice that would provide qualified veterans a reasonable opportunity to claim their preference points. The type and form of notice to be given is a matter that should *170be determined on remand. In opposing any requirement that they make any adjustment in rankings on currently used pre-1985 lists, the City defendants have argued that, “[u]pon information and belief, at any given time, there are hundreds of eligible lists with literally thousands and thousands of new applicants and eligibles. It would require countless additional employees and enormous expense to the City to even attempt to ascertain who might be eligible for retroactive relief.” (City defendants’ September 25, 1987 letter to this Court.) Conclusory assertions made “upon information and belief,” however, are a wholly inappropriate foundation on which to allow the City defendants to continue to give effect to provisions that have been declared unconstitutional. The record provides no factual basis for assessing the practical difficulties in providing some form of notice that would provide a reasonable opportunity for qualified veterans, currently ranked on a list compiled using the unconstitutional provisions, to claim their preference points. For example, it is unclear how many pre-March 1985 eligibility lists remain in current use; the City defendants themselves indicate that “many” of the pre-March 1985 lists have expired and, under state law, cannot be revived. Nor does the record give any indication of when the current pre-March 1985 lists are due to expire, or how many persons are on those lists, or the manner in which the “new applicants” referred to are processed. Also unanswered are questions such as whether the City has any way to determine which applicants are veterans, whether it has a convenient method of communicating directly with all persons on the lists, and whether communications already required could be used to give effective notice of the rights declared in this litigation, see, e.g., N.Y.Civ. Serv.Law § 61.3 (McKinney Supp. 1988) (requiring written notice to certain certified candidates on an eligibility list whenever another candidate is appointed to a position).

These are among the matters to be explored on remand. Their assessment will influence the manner in which defendants will be required to give notice of their constitutional obligations but should not prevent the granting of an injunction that, at a minimum, conforms to the guidelines set out at the beginning of this section.

C. Individual Relief

In our 1985 opinion, we remanded to the district court for further proceedings regarding issues relating to plaintiffs’ requests for individual relief and instructed the court to consider “whether either plaintiff should receive immediate appointment, if so to what position, and whether awards of backpay or retroactive seniority are appropriate, as well as issues relating to defendants’ defenses of immunity.” Soto-Lopez I, 755 F.2d at 281. We also noted that the City defendants had failed to controvert the factual statements presented by plaintiffs in their statements pursuant to Local Rule 3(g), which plaintiffs contend establish their eligibility for the positions they sought; but we concluded that the City should be given another opportunity to comply with that Rule in order to demonstrate any genuine issue as to either plaintiff’s eligibility. Id. On the remand, none of the reasonably anticipated events occurred.

First, the City defendants did not take advantage of the renewed opportunity to present a Rule 3(g) statement and evidence that either plaintiff was not qualified for bonus points or appointment. Though the City defendants stated in a letter to the court that one of the plaintiffs had failed to prove that he met the eligibility requirements for the civil service position sought, the record does not indicate that defendants made any presentation designed to satisfy Fed.R.Civ.P. 56(e) in opposition to plaintiffs’ supported motion for summary judgment. Instead, the City defendants submitted to the district court a proposed order that required them to grant each plaintiff his veterans’ bonus points if he met the requirements of § 85 and to appoint him if he established that he was qualified for the position sought.

For their part, plaintiffs failed to propose any decretal provisions that finally adjudicated their individual claims. Rather, their *171.first proposed provision for individual relief required the City defendants to grant them veterans’ preference points “if” they met “the requirements of Section 85 of the New York Civil Service Law other than” the past-residency requirements. The two succeeding paragraphs of plaintiffs’ proposed order were similarly conditional, ordering that the City defendants were to appoint or promote the plaintiffs “if” with the veterans’ bonus points they would have been appointed or promoted from the designated eligibility lists.

Lastly, the district court did not decide the entitlement of either plaintiff to either an award of veterans’ preference points or an immediate appointment. Instead, it entered the 1987 Judgment ordering the City defendants to award each plaintiff veterans’ bonus points “if” he met the requirements of § 85 other than past-residency, and it ordered defendants to appoint or promote each plaintiff “provided he is otherwise qualified for the position.” This did not finally adjudicate the claims for individual relief.

Accordingly, we remand once again for the district court to decide to what individual relief, if any, each plaintiff is entitled. With respect to this adjudication, we leave it to the discretion of the district court to determine whether the City defendants are to be given a third chance to controvert the Rule 3(g) statements submitted by plaintiffs in support of their motion for summary judgment. If the district court determines that either plaintiff was entitled to be appointed or promoted, the court will need to decide that plaintiff’s claim for backpay, including the immunity defense raised by the City defendants.

CONCLUSION

Three years have passed since this Court declared the past-residency requirements unconstitutional and l-Vfe years have passed since the Supreme Court affirmed. In the interim, the City defendants apparently have appointed many persons to positions that, for all we know, would have gone to veterans who would have ranked ahead of the appointees had the veterans not wrongfully been denied bonus points. Defendants’ submissions to both the district court and this Court make it plain that unless enjoined, the City defendants have no intention of ceasing to use current lists that reflect the unconstitutional provisions. Such continued use should be enjoined in accordance with the guidelines we have set forth in Part II.B. above. We trust that the district judge’s calendar will permit the expeditious resolution of all of the remaining issues necessary for the fashioning of appropriate injunctive relief.

Except to the extent that the 1987 Judgment declares the past-residency requirements unconstitutional, that Judgment is vacated, and the case is remanded for further proceedings not inconsistent with this opinion.

Costs to plaintiffs.