  The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov
systems of semilinear partial differential equations \label{GDMZabstract}
\frac{\partial^2 u}{\partial x_i\partial x_j}=f_{ij}\Big(x_k,u,\frac{\partial
u}{\partial x_l}\Big), 1\leq i<j\leq n, k,l\in\{1,...,n\} for a real-valued
function $u(x_1,...,x_n)$ are studied with particular reference to the linear
systems in this equation class.
  System (\ref{GDMZabstract}) will not generally be involutive in the sense of
Cartan: its coefficients will be constrained by complicated nonlinear
integrability conditions. We derive geometric tools for explicitly constructing
involutive systems of the form (\ref{GDMZabstract}), essentially solving the
integrability conditions. Specializing to the linear case provides us with a
novel way of viewing and solving the multi-dimensional $n$-wave resonant
interaction system and its modified version as well as constructing new
examples of semi-Hamiltonian systems of hydrodynamic type. The general theory
is illustrated by a study of these applications.
