About 2,000 U.S. veterans of the Persian Gulf War have filed a $1 billion civil suit against 20 prominent German companies, alleging that the firms helped Iraqi leader Saddam Hussein build a poison-gas arsenal. The German magazine Stern reports the action in a story to be published Thursday. The suit, Stern says, seeks $1 billion in damages for exposure to chemical weapons they believe left them with "Gulf War Syndrome," a mysterious collection of symptoms that includes chronic fatigue, headaches, rashes and painful limbs. Some of the German companies today called the suit groundless; no German firm has ever acknowledged selling Hussein materials that were used to make chemical weapons. TIME Defense correspondent Mark Thompson says the vets will have a tough time proving any cause-and-effect in court when the Pentagon itself believes the illnesses have no single cause. Still, Thompson adds, Germany was far less strict than the U.S. about selling multiple-use materials to Iraq in the past, and "whether it's tanks, airplanes or ingredients for poison gas, the bad guys in that drama are clearly the Europeans."