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

We show that matrix $Q\times Q$ Self-dual type $S$-integrable Partial Differential Equations (PDEs) possess a family of lower-dimensional reductions represented by the matrix $ Q \times n_0 Q$ quasilinear first order PDEs solved in \cite{SZ1} 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.