In this paper we solve the following problems: (i) find two differential operatorsP andQ satisfying [P, Q]=P, whereP flows according to the KP hierarchy ϖP/ϖt n =[(P n/p )+,P], withp:=ordP≥2; (ii) find a matrix a integral representation for the associated τ-function. First we construct an infinite dimensional spaceW= spanℂ{ψ 0(z,ψ 1(z,...)} of functions ofzεℂ invariant under the action of two operators, multiplication byz p andA c :=zϖ/ϖz−z+c. This requirement is satisfied, for arbitraryp, ifψ 0 is a certain function generalizing the classical Hänkel function (forp=2); our representation of the generalized Hänkel function as adouble Laplace transform of a simple function, which was unknown even for thep=2 case, enables us to represent the τ-function associated with the KP time evolution of the spaceW 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 contourγ≔γ -+γ - ⊂ℂ defined byγ ± = ℝ+e±πi/p. The new integrals above relate to matrix Laplace transforms, in contrast with matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P, Q]=1.