In this paper we investigate the Riemann boundary problem $$ \Phi^+(t)=G(t)\Phi^-(t)+g(t) $$for$n$pairs of functions. The solutions$\Phi^\pm$are to belong to the classes$E_p^\pm$; the given function g belongs to the class$L_p$$(1<p<\infty)$. We enlarge the class of coefficients$G$for which the Noether theory remains valid. In the case$n=1$,$p=2$, necessary and sufficient conditions for Noetherianness are obtained. It is shown that the class of matrix-functions which admit factorization coincides with the class for which the Noether theory is valid. In the case$n=1$it is shown that one of the defect numbers is zero.