As the United States and its major allies were gearing up for a no-fly zone over Libya, a senior U.S. general was asked over a private dinner whether the strained American military was up for another Middle East conflict. We could easily impose a no-fly zone, he replied, according to one of his tablemates, but what would the objective be?

More than a month and hundreds of coalition airstrikes later, the answer has remained elusive, and the war goes on.

Under the terms of U.N. Security Council Resolution 1973, which on March 17 authorized Western air operations over Libya, NATO aircraft are bombing to protect civilian lives. But according to increasingly explicit statements from European leaders, they are also deployed to help an armed rebellion defend its positions and pressure Moammar Gaddafi and his sons to give up power in Tripoli.

The equivocation, according to observers inside and outside the alliance, has fostered frustration in European capitals at what seems increasingly to be a stalemated ground war along the sandy expanses of Libya’s Mediterranean shore. Moreover, it has strained the cohesion of NATO’s 28 member countries, some of which insist on sticking strictly to the civilian protection mission while others say the only way to protect Libya’s population is to get rid of Gaddafi.

President Obama’s reluctance to stay fully involved has been perceived by some European officials as a fraying of NATO’s solidarity principle. In a joint declaration last week, Obama joined French President Nicolas Sarkozy and British Prime Minister David Cameron in declaring that Gaddafi must go, a gesture toward their expansive war goals. But, Europeans complained, he remained unwilling to fully commit U.S. resources to help make that happen.

Perhaps the most debilitating development since March 31, when the United States transferred command of the Libyan operation to NATO, has been the absence of U.S. military leadership. This has left the Naples-based multinational NATO command for Operation Unified Protection without a naturally dominant voice.

The frustration has been particularly visible in France and Britain, the two countries whose leaders pushed hardest for Western intervention and who have been at the forefront in seeking a victory for the rebels. By their yardstick — helping rebel forces topple Gaddafi — the bombing campaign has fallen short.

“According to the logic of what they say, it can’t be over until Gaddafi is out of there,” said Robert Danin, a Middle East specialist at the Council on Foreign Relations. He noted that “questions, contradictions and ambiguities” have plagued the operation from its outset.

Gaddafi has survived five weeks of punishing airstrikes, and his military has not yet betrayed him, as officials in Paris and London were hoping. In a notable display of candor, French Foreign Minister Alain Juppe suggested this week that alliance leaders — including his boss, Sarkozy — may have underestimated Gaddafi’s staying power in deciding to go to war.

“When the decision to take action was taken, there was a feeling that, in the wake of what happened in Tunisia and Egypt, this would be relatively quick,” Danin recalled. “Instead, it is turning out to be a protracted civil war on the ground.”

Rebel forces in eastern Libya have proved unable to maintain advances from their Benghazi headquarters and forward positions at the crossroads town of Ajdabiya. Their major prize in western Libya, Misurata, has come under relentless barrages from Libyan army artillery and rocket launchers, leading the town council to plead for intervention by foreign ground troops.

Measured by the situation in Misurata, NATO bombing has also fallen short in protecting civilians during the most recent combat. Asked Tuesday about the humanitarian crisis there, Brig. Gen Mark van Uhm, a Dutch national who is NATO’s chief of operations, noted that air power alone cannot guarantee protection for civilians in Misurata. Bombers are limited in what they can do, he said at a briefing here, because of a need to avoid collateral casualties in what has become close-quarters urban warfare.

In response to a growing impression that the rebellion has bogged down, Cameron on Tuesday dispatched a dozen British military advisers to help rebel leaders in Benghazi better organize their forces. France and Italy followed up the next day with separate announcements that they, too, will send in military advisers — on a national basis beyond NATO’s purview — to work in tandem with the British.

“We will help you,” Sarkozy said Wednesday in a meeting with the senior rebel leader, Mustafa Abdel Jalil, according to a Sarkozy aide.

Despite their goals, Cameron and Sarkozy have disavowed any plans to send in ground troops to fight alongside the rebels. “That’s not what we want, that’s not what the Libyans want, that’s not what the world wants,” Cameron told a BBC interviewer Thursday.

But the commitment of military advisory teams, inching together toward deeper involvement, was taken as a measure of impatience with the way the conflict is playing out.

The decision to act outside the NATO umbrella suggested the three countries thought they could not get a consensus within the alliance for sending military advisers, since that seems to go beyond the authorizing U.N. resolution. In that light, French Defense Minister Gerard Longuet suggested that another, broader resolution may be necessary before the conflict is over.

Other NATO countries have proved less enthusiastic. Germany, a key ally, has refused to participate at all in the Libyan operation, saying military intervention is not the solution to the country’s problems. Italy, another NATO ally with a strong military, has offered air bases but refused NATO entreaties to participate in the bombing, citing its colonial past in Libya.

In seemingly contradictory decisions, however, Italy has dispatched relatively large numbers of advisers to Benghazi, in addition to joining France and Britain in promising a military team to help the rebels set up a more cohesive command structure.

Under diplomatic pressure, Norway, Canada, Denmark and Belgium assigned aircraft to the bombing campaign, but not on the scale of Britain and France. Despite repeated NATO appeals, a senior NATO diplomat said, the Netherlands and Spain authorized their pilots only to patrol the no-fly zone, along with Sweden, Qatar, Jordan and the United Arab Emirates from outside the alliance. Turkey, another alliance member reluctant to get involved in the bombing, sent aircraft only to help enforce a U.N. arms embargo.

The largest NATO country withholding assets, however, is the United States. After dominating the initial bombing and cruise missile strikes, it turned over command to NATO and ceased most bombing missions March 31. Since then, U.S. planes have been flying reconnaissance and command-and-control missions but launching airstrikes only at static air defense installations.

The Obama administration refused a request from France last week to return to the campaign with the full force of its air and naval fleets, which are unmatched by European arsenals. Some observers have lamented in particular the absence of U.S. A-10 Warthog ground-attack jets — specifically designed for close air support — and AC-130 turboprop gunships. In fact, the low- and slow-flying planes were hardly deployed even when the United States was fully engaged, because of fears they would be shot down by the Libyan army’s shoulder-fired antiaircraft missiles.

NATO officials insisted that the number of strike missions has not abated under their command. But they declined to say how many planes assigned to carry out strike missions actually drop bombs or fire missiles before returning to base. Rebel leaders have complained that the number of fired munitions fell dramatically, and the time required to call in an airstrike was multiplied several times, as soon as U.S. commanders left the controls.