London, England (CNN) -- A few dozen test flights Sunday offered hope that the skies over much of Europe may be safe for air travel, but officials made no promises that the massive disruptions due to volcanic ash are about to go away.
"The results coming from these flights is... there's no impact in the area," European Union Secretary of State Diego Lopez Garrido said.
EU Transport Commissioner Siim Kallas said if the ash cloud continues "moving as it moves, then tomorrow almost 50 percent of European [Union] space will be risk free." That would allow more flights to resume, he said. "But we'll see [Monday] what the picture shows."
Two key air travel groups issued a joint statement pushing authorities to ease flying restrictions. Airports Council International (ACI) Europe, which represents airports, and the Association of European Airlines (AEA) said they question "the proportionality of the flight restrictions currently imposed."
But an expert who has flown into the skies to check conditions said he believes it will be "a few days yet" before it's safe to fly.
A spokeswoman for KLM, one of the airlines that conducted test flights, told CNN the flights show European airspace is safe, with the exception of Iceland. CNN's Gary Tuchman in Iceland reported that there were airspace problems in eastern Iceland. In the capital, Reykjavik, flights were taking off and landing.
British Airways, which conducted a test flight Sunday, said "conditions were perfect and the aircraft encountered no difficulties." The plane will undergo "a full technical analysis" next.
European transport ministers plan to discuss the results of flight tests at a technical meeting Monday.
The Italian Civil Aviation Authority (ENAC) announced it will re-open the entire Italian air space to all flights Monday.
The British government said British airspace will be closed for at least another 24 hours.
British Transport Secretary Andrew Adonis said officials were working around the clock to establish whether safe flight paths could be identified.
"Urgent discussions are taking place with European and international regulatory agencies. We want to be able to resume flights as soon as possible, but safety remains my paramount concern," Adonis said.
At London's Heathrow airport, a crowd broke out into cheers as a flight took off to test the skies.
Since the eruption beneath southern Iceland's Eyjafjallajokull glacier worsened last week, prompting local evacuations and affecting European airspace, airlines have been losing at least $200 million a day, according to the International Air Transport Association, the trade group representing airlines.
Olivier Jankovec, director general of ACI Europe, said airports have lost close to 136 million euros ($184 million U.S.) so far. More than 6.8 million passengers have been affected, he said in a statement, adding that the effect is worse than after the September 11, 2001, terrorist attacks on the United States.
"While safety remains a non-negotiable priority, it is not incompatible with our legitimate request to reconsider the present restrictions," he said.
"While Europe's airlines and airports consider safety to be an absolute priority, they are questioning the proportionality of the flight restrictions currently imposed," ACI Europe and the AEA said in their joint statement. "The eruption of the Icelandic volcano is not an unprecedented event and the procedures applied in other parts of the world for volcanic eruptions do not appear to require the kind of restrictions that are presently being imposed in Europe."
But an expert who has flown over Europe to check the air said he saw "dangerous" conditions.
Guy Gratton, head of the Facility for Airborne Atmospheric Measurements at Britain's Cranfield University, flew into the skies Thursday and saw "a really strange and complex set of layers of ash," with a layer of perfectly clear air suddenly giving way to a layer of ash, he told CNN. If particles of ash enter a jet engine, when they come out they can solidify on turbine blades, he said.
A group of his colleagues took to the skies Sunday, and in some places saw "quite high concentrations of ash," he said.
"I suspect it's going to be a few days yet" before it's safe to fly, Gratton added.
Across much of Europe, huge crowds have packed into airports, train stations and ferry lines, desperate for a way to get to their destinations -- which, in many cases, is home.
"My heart is aching," Jen Patterson told "CNN Sunday Morning." She and her husband, Steve, were traveling in the Netherlands when the problems began. They've been unable to get home to their four children, all younger than 9, who are being cared for by friends and family.
An airline Sunday booked them on a flight Friday from Madrid to Dulles airport near Washington. "So the next challenge is getting to Madrid, whether by plane or by car," said Steve Patterson.
In various ways, millions of people have been affected by the aviation disruption, which some officials call worse than after the September 11, 2001 terrorist attacks on the United States.
Some European airports reopened Sunday, including several in France and Germany, and all 16 that had been closed in Spain. But officials in each country emphasized that decisions were being made around the clock and could change at any time.
Airports in much of the continent remained closed, with passengers unable to go anywhere. Many hotels quickly ran out of rooms.
As the situation became more dire, Britain said Sunday it was looking at whether to draft in the Royal Navy to help those stranded. The government planned efforts overnight to see what military and commercial vessels were available for possible deployment.