On extremal decomposition problem in the family of general type systems of domains

We prove a theorem extending results of the theory of extremal decomposition problems to families of systems of domains of general type. The considered families of systems of domains contain domains similar in the small to end and strip domains of a quadratic differential having poles of arbitrary orders $n_k\ge3$ at some marked points $c_k$, $k=1,\dots,p$. In this case, we give a simple definition of reduced modules for the considered systems of domains. Some other definitions for the treated systems of domains are also considered. Some examples are given illustrating the possibilities of applications of the theorem obtained in the problems on extremal decomposition.