  In this paper we construct nonlinear partial differential equations in more
than 3 independent variables, possessing a manifold of analytic solutions with
high, but not full, dimensionality. For this reason we call them ``partially
integrable''. Such a construction is achieved using a suitable modification of
the classical dressing scheme, consisting in assuming that the kernel of the
basic integral operator of the dressing formalism be nontrivial. This new
hypothesis leads to the construction of: 1) a linear system of compatible
spectral problems for the solution $U(\lambda;x)$ of the integral equation in 3
independent variables each (while the usual dressing method generates spectral
problems in 1 or 2 dimensions); 2) a system of nonlinear partial differential
equations in $n$ dimensions ($n>3$), possessing a manifold of analytic
solutions of dimension ($n-2$), which includes one largely arbitrary relation
among the fields. These nonlinear equations can also contain an arbitrary
forcing.
