The bunch of differential operators generated by the differential expression of the second order whose main characteristic polynomial has one root with the multiplicity two is considered, when the coefficients of differential expression contain only positive Fourier index in the space $L_2(-\infty,\infty)$. The solutions of corresponding differential equations are constructed. It is obtained that the bunch has purely continuous spectrum coinciding with whole real axis. For other points of complex plane of spectral parameter the bunch resolvent is integral operator with Carleman type kernel. The decomposition by main functions of continuous spectrum is obtained for triply continuous differentialble compactly supported functions.