  In this paper we solve the following problems: (i) find two differential
operators P and Q satisfying [P,Q]=P, where P flows according to the KP
hierarchy \partial P/\partial t_n = [(P^{n/p})_+,P], with p := \ord P\ge 2;
(ii) find a matrix integral representation for the associated $\t au$-function.
First we construct an infinite dimensional space {\cal W}=\Span_\BC
\{\psi_0(z),\psi_1(z),... \} of functions of z\in\BC invariant under the action
of two operators, multiplication by z^p and A_c:= z \partial/\partial z - z +
c. This requirement is satisfied, for arbitrary p, if \psi_0 is a certain
function generalizing the classical H\"ankel function (for p=2); our
representation of the generalized H\"ankel function as a double Laplace
transform of a simple function, which was unknown even for the p=2 case,
enables us to represent the \tau-function associated with the KP time evolution
of the space \cal W as a ``double matrix Laplace transform'' in two different
ways. One representation involves an integration over the space of matrices
whose spectrum belongs to a wedge-shaped contour \gamma := \gamma^+ + \gamma^-
\subset\BC defined by \gamma^\pm=\BR_+\E^{\pm\pi\I/p}. The new integrals above
relate to the matrix Laplace transforms, in contrast with the matrix Fourier
transforms, which generalize the Kontsevich integrals and solve the operator
equation [P,Q]=1.
