Summary: This paper contains some results of investigation of the internal smoothness of the solution for one mixed problem for an evolution system with degeneration. This problem consists of an evolutional system with second derivative in time, degenerating into an elliptic system on the initial plane, homogeneous Dirichlet boundary, and some initial conditions. The system contains, in particular, some classes of parabolic and hyperbolic systems. The problem is considered in a cylindrical domain.