An internal-combustion engine system includes an internal-combustion engine with a pertaining engine exhaust gas system, an emissions control device having an NO.sub.x storage catalyst arranged in the exhaust gas system, a lambda probe for detecting the engine air ratio and devices for the periodic desulfurization of the catalyst at a raised temperature and a rich air ratio of the latter, as well as to an operating process suitable for this system. The desulfurization devices comprise secondary air feeding devices for feeding secondary air into the NO.sub.x storage catalyst. During the desulfurization phases, the engine is operated at a rich air ratio and secondary air is metered into the catalyst.