More flights, lower fares.

That was the promise of the so-called open skies agreement, which, at the end of this month, will allow more airlines to fly between the U.S. and Europe.

The reality: more flights, mixed fares. If you plan to fly to Europe in the next couple of months, you'll find some good prices. But summer fares so far are running more than $1,000 round trip from LAX to London and more than $1,400 to Paris or Rome, with taxes and fees.

What's a traveler to do? Go before June, if you can, when transatlantic flights cost hundreds less. And be patient: Sales on summer flights may start soon, if the past is any indicator.

"If you're smart, you'll find bargains," said Tom Parsons, chief executive of the BestFares.com travel website.

Here are five keys to saving on European flights:

1. Shop spring sales: This year's deals are beating last year's, several experts say.

"Right now, you've got some bargain-basement deals for Europe in April and May," Parsons said.

Round trips between LAX and Paris in spring have been going for as little as $498 plus taxes and fees, he said, and London-LAX round trips have been even cheaper.

Overall, the lowest fares published last week for spring travel from Los Angeles and New York to London's Heathrow Airport, and from Los Angeles to Paris' Charles de Gaulle, were more than 20% below fares published this time last year, said Bob Harrell, of Harrell Associates, a New York-based travel and aviation consulting firm.

Summer travel is another matter. Although the lowest LAX-Heathrow summer fare last week was well below last year's, LAX-De Gaulle was up nearly 6% and New York-Heathrow was unchanged, Harrell found.

"Usually the peak-season fares start pretty high," he said. "They're trying to see if the suckers will pay them."

If they don't, we'll see summer flights go on sale, probably in April, he said. If they do, we won't. Harrell is betting on lower prices.

2. Get savvy about open skies. Last year the U.S. and the European Union agreed to loosen restrictions on the transatlantic air market, allowing U.S. and European carriers to fly nonstop from any city in the U.S. to anywhere in the EU.

That means transatlantic fliers are getting more choices as airlines add routes from U.S. cities. On March 30, for instance, Air France will start flying to London's Heathrow from LAX. Overall, airlines have scheduled 8% more flights between the U.S. and Europe from April through June than in the same period last year, according to the Airline Planning Group and eSkyGuide, which publishes flight information.

More choice is not always bringing lower prices. That's partly because transatlantic flights continue to be popular, said Stanley Gyoshev, co-founder of Lessno.com, a bargain-fares website. And with high fuel costs and the weak U.S. dollar, there's not much room for fares to fall, he added.

Still, discounts happen, if you know when and where to look.

3. Watch for new routes: When an airline adds service, it usually offers low fares at first. Air France last month held a two-day sale with round trips for $262 plus taxes and fees on its new LAX-London route. Competitors often match introductory fares, which may be so low that it may be cheaper to fly to Europe through a newly added U.S. gateway than to fly from your home city.

"Be creative," Parsons said.

4. Fly to London first: Heathrow is Europe's biggest hub, and with open skies, competition is heating up there the fastest. In April, airlines will fly 21% more seats between the U.S. and Heathrow than in April last year, according to the Airline Planning Group and eSkyGuide.

So rather than fly to Paris or Rome, which often command hundreds more for nonstop flights, you may save by flying to London and connecting to those cities and others on low-cost local carriers.

5. Compare code-shares: Under so-called code-share agreements, which are common in the industry, airlines sell tickets on each other's flights. And they don't always charge the same fares. The difference can amount to hundreds of dollars on a transatlantic flight.

"Look at every fare," Parsons advised.

jane.engle@latimes.com