The subject of the paper is the solubility of the Cauchy problem for strictly hyperbolic systems of Monge-Ampère equations and, in particular, for quasilinear systems of equations with two independent variables. It is proved that this problem has a unique maximal solution in the class of immersed many-valued solutions. Maximal many-valued solutions have the following characteristic property of completeness: either the characteristics of distinct families starting at two fixed points in the initial curve in the compatible directions intersect or the lengths of the characteristics in either family starting in the same direction from the interval of the initial curve connecting the fixed points make up an unbounded set. The completeness property is an analogue of the property that a non-extendable integral curve of an ordinary differential equation approaches the boundary of the definition domain of the equation.