  We represent a version of multidimensional quasilinear partial differential
equation (PDE) together with large manifold of particular solutions given in an
integral form. The dimensionality of constructed PDE can be arbitrary. We call
it the $n$-wave type PDE, although the structure of its nonlinearity differs
from that of the classical completely integrable (2+1)-dimensional $n$-wave
equation. The richness of solution space to such a PDE is characterized by a
set of arbitrary functions of several variables. However, this richness is not
enough to provide the complete integrability, which is shown explicitly. We
describe a class of multi-solitary wave solutions in details. Among examples of
explicit particular solutions, we represent a lump-lattice solution depending
on five independent variables. In Appendix, as an important supplemental
material, we show that our nonlinear PDE is reducible from the more general
multidimensional PDE which can be derived using the dressing method based on
the linear integral equation with the kernel of a special type (a modification
of the $\bar\partial$-problem). The dressing algorithm gives us a key for
construction of higher order PDEs, although they are not discussed in this
paper.
