  We develop a new integration technique allowing one to construct a rich
manifold of particular solutions to multidimensional generalizations of
classical $C$- and $S$-integrable Partial Differential Equations (PDEs).
Generalizations of (1+1)-dimensional $C$-integrable and (2+1)-dimensional
$S$-integrable $N$-wave equations are derived among examples. Examples of
multidimensional second order PDEs are represented as well.
