A burner system includes at least one radiant heating tube (22) and a first regenerator (48) disposed at a first end (24) of the tube. A second regenerator (50) is disposed at a second end (26) of the radiant heating tube (22). The first regenerator (48) and the second regenerator (50) are connected to a valve system (54) having first and second operating states for alternately supplying the radiant heating tube (22) with combustion air via one regenerator (48, 50) and for discharging exhaust gases via the other regenerator (48, 50). At least one inner tube (34) is disposed inside and extending along the radiant heating tube (22) at least in sections. The inner tube (34) is connected to a fuel supply line (76) and has outlet openings (46) provided along the longitudinal extension of the inner tube (34).