For integrable partial differential systems which are dispersionless a.k.a. hydrodynamic-type, i.e. can be written as quasilinear first-order homogeneous systems, in the case of two independent variables see e.g. these lecture notes, Section 3 of this article, and references therein; in the case of three independent variables see Section 3 of this same article, subsubsection 10.3.3 of this book, and these lecture notes, and references therein; for the case of four independent variables see introductory part of this article, the same book as before, and references therein. The book in question in fact is a good introduction to integrable partial differential systems in general.