Boeing plans to keep building its flagship jetliner while engineers try to solve battery problems that have grounded most of the 787 fleet.

It’s not clear how long the investigation — or the fix — will take. But it won’t be cheap for Boeing or for the airlines that had sought the prestige of flying the world’s most sophisticated passenger plane — a marvel of aviation technology that right now can’t even leave the tarmac, let alone cross continents and oceans.

Boeing’s newest jet was grounded worldwide Thursday after one experienced a battery fire and another had to make an emergency landing because pilots smelled something burning. Airlines and regulators canceled all Dreamliner flights.

The groundings were a sign of how seriously regulators take any threat of an in-flight fire. National Transportation Safety Board photos of the battery container from a Jan. 7 fire on a Japan Airlines plane showed a blue box with black smudges and blackened wiring and batteries inside.

LOT Polish Airlines suffered the highest-profile embarrassment of any of Boeing’s customers late Wednesday, just as it was showing off new service between Warsaw and Chicago.

The plane’s captain learned of the FAA grounding order while making the inaugural flight from Warsaw to Chicago. The airline canceled the return trip — and a ceremony at O’Hare Airport that was to include airline officials and Mayor Rahm Emanuel. Passengers who were eager to ride the airline’s first flight back to Warsaw had to look for a hotel room instead.

The airline said Thursday that it might seek compensation from Boeing for the grounding of its two 787s.

Boeing currently builds five 787s per month. It hasn’t delivered any since Jan. 3, before the plane started experiencing a spate of problems that also included fuel and oil leaks, a cracked cockpit window and a computer glitch that erroneously indicated a brake problem.

Boeing spokeswoman Lori Gunter said no deliveries had been scheduled during that time.

She would not discuss planned deliveries.

Regardless of delivery schedules, it’s cheaper for Boeing to build the planes and then go back and fix them than it is to shut down production.

All Nippon Airways said its 18th 787 is due at the end of this month, but it won’t take delivery until 787 flights resume.

The FAA grounding was a stunning setback for Boeing. The plane has been in the works since 2003, a time when modern jetliners were built of aluminum and powered many of their internal systems with incoming air from outside the plane. Boeing engineers figured they could get better fuel efficiency by making the plane out of carbon composites, a sort of lightweight, high-tech plastic. And they used electricity rather than air because it saved space and weight.

The 787 was tested extensively both before and after its first test flight in 2009. The FAA said its technical experts logged 200,000 hours testing and reviewing the plane’s design before it was certified in August 2011.

Six test planes ran up about 4,645 flight hours. About a quarter of those hours were flown by FAA flight test crews, the agency said in 2011.

New 787s sell for more than $200 million at list prices. For that kind of money, airline customers get warranties and in some cases a promise from Boeing to cover costs if the plane is grounded.