On periodic in the plane solutions of second order linear hyperbolic systems

summary:Sufficient conditions for the problem \[ {\partial ^2 u\over \partial x\partial y}=P_0(x,y)u+ P_1(x,y){\partial u\over \partial x}+P_2(x,y){\partial u\over \partial y}+ q(x,y), u(x+\omega _1,y)=u(x,y),\quad u(x,y+\omega _2)=u(x,y) \] to have the Fredholm property and to be uniquely solvable are established, where $\omega _1$ and $\omega _2$ are positive constants and $P_j:R^2\rightarrow R^{n\times n}$ $(j=0,1,2)$ and $q:R^2\rightarrow R^n$ are continuous matrix and vector functions periodic in $x$ and $y$