Matrix equations of hydrodynamic type as lower - dimensional reductions of S - integrable systems

We show that any classical matrix Q × Q S-integrable Partial Differential Equation (PDE) obtainable as commutativity condition of a certain type of vector fields possesses a family of lower-dimensional reductions represented by the matrix Q × n 0 Q quasilinear first order PDEs solved in [29] by the method of characteristics. In turn, these PDEs admit two types of available particular solutions: (a) explicit solutions and (b) solutions described implicitly by a system of non-differential equations. The later solutions, in particular, exhibit the wave profile breaking. Only first type of solutions is available for (1+1)-dimensional nonlinear S-integrable PDEs. (1+1)-dimensional N-wave equation, (2+1)-and (3+1)-dimensional Pohlmeyer equations are represented as examples. We also represent a new version of the dressing method which supplies both classical solutions and solutions with wave profile breaking to the above S-integrable PDEs.