Periodic systems for the higher-dimensional Laplace transformation

$ y_{,k\l}+a_{k\l}^ky_{,k}+a_{k\l}^\ly_{,\l}+c_{k\l}y=0\ , \quad 1\le k\ne\l\le n\ , $

where the coefficients are smooth functions satisfying certain integrability conditions. Generalizing the classical theory of second order linear hyperbolic partial differential equation in the plane, we consider higher-dimensional Laplace invariants of a system of the above class. These invariants are characterized as functions which must satisfy a set of differential equations. We establish a normal form for any system of the above class in terms of these invariants. Moreover, we solve the periodicity problem for the higher-dimensional Laplace transformation applied to such systems, generalizing a classical theorem of Darboux which shows that for $n=2$ a 1-periodic equation is equivalent to the Klein-Gordon equation.