Saratov State University, Russia, 410012, Saratov, Astrahanskaya st., 83, KornevVV@info.sgu.ru, KhromovAP@info.sgu.ruTheobjectofthepaperisDiracsystemwithantiperiodicboundaryconditionsandcomplex-valuedconditionspotential.Anewmethodis suggested for investigating spectral properties of this boundary problem. The method is based on the formulas of the transformoperators type. It is rather elementary and simple. Using this method asymptotic behaviour of eigenvalues is specificated and it isproved that eigen and associated functions form Riesz basis with brackets in the space of quadratic summerable two-dimensionalvector-functions since eigenvalues may be multiple. The structure of Riesz projection operators is also studied. The results of thepaper can be used in spectral problems for equations with partial derivatives of the 1-st order containing involution.Key words: Dirac system, spectrum, asymptotics, Riesz basis.