We introduce the notion of an $L$-Ponomarev system $(f,M,X,\mathcal P_n^*)$, and give characterizations of certain msss-images (resp., mssc-images) of locally separable metric spaces. As an application, we get a new characterization of quotient msss-images (mssc-images) of locally separable metric spaces, which is helpful in solving Velichko's question (1987).