Lufthansa today called for new tests by scientists on the volcanic ash cloud currently preventing flying over much of Europe.
The airline said it believed new tests would show it was safe to fly.
The call comes after the German national carrier staged flights at various altitudes to see what effect the ash cloud would have on its planes.
A spokesman for the airline said the tests including flying at the low altitude of 3,000 metres which some experts have suggested is a possible way of avoiding the ash cloud.
"We had no problems with our flights. Everything went perfectly. There were no problems with the cockpit windows or with the fuselage which we were told could de damaged by the cloud," he said.
"Our planes were imperfect condition afterwards."
He said the airline now wanted new tests by scientists as current "mathematical models were not in correlation with what is happening in the skies".
New tests would enable flights to be resumed as soon as possible, he said.
The ash is from the Eyjafjallajoekull volcano in south east Iceland.
Aviation experts have warned that particles of rock, glass and sand in the ash cloud could jam airline engines.
All four engines on a BA plane shut down when it flew into an ash cloud over Indonesia in 1982. The plane landed safety after the pilots got the engines to restart.
KLM said it had "successfully" carried out ten test flights over the weekend to see whether it was safe to fly in the volcanic ash cloud.
The first flight, on Saturday, "encountered no problems whatsoever" enabling a further nine flights to go ahead.
The Dutch airline said that "none of the test flights operated to date has encountered any problem.
Peter Hartman, the airline's president and ceo, said: "It is completely safe to operate flights during hours of daylight".
KLM said it hoped to receive permission to resume part of its operations as soon as possible."
Air France said it had also successfully operated the first of a series of test flights from Paris-Charles de Gaulle to Toulouse-Blagnac airport.
"Conditions during the flight were normal and no damage was reported following an initial visual inspection. More in-depth inspections are ongoing," the airline said.
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