The Riemann matrix for some systems of the differential hyperbolic-type equations of the high order

Solutions to some boundary value problems for systems of hyperbolic partial differential equations can be constructed explicitly in terms of the Riemann matrix. In this regard, the question of explicitly constructing the Riemann matrix for high-order hyperbolic systems of equations is relevant.
We consider a system of third-order hyperbolic partial differential equations with three independent variables. For the specified system, the Riemann matrix is constructed as a solution to a special Goursat problem. Furthermore, the Riemann matrix satisfies a Volterra integral equation. The Riemann matrix is expressed explicitly in terms of a hypergeometric function of a matrix argument. Similarly, a system of fourth-order hyperbolic partial differential equations with four independent variables is considered. These results are generalized for a system of hyperbolic partial differential equations of order $n$ that does not contain derivatives of order less than $n$.