  We study factorizations of operator valued functions of weighted Schur
classes over multiply-connected domains. There is a correspondence between
functions from weighted Schur classes and so-called ``conservative curved''
systems introduced in the paper. We show that the fundamental relationship
between invariant subspaces of the main operator of a conservative system and
factorizations of the corresponding operator valued function of Schur class,
which is well known in the case of the unit disk, can be extended to our case.
We develop new notions and constructions and discuss changes that should be
made to the standard theory to obtain desired generalization.
