SQUARED EIGENFUNCTIONS OF THE (MODIFIED) KP HIERARCHY AND SCATTERING PROBLEMS OF LOEWNER TYPE

It is shown that products of eigenfunctions and (integrated) adjoint e igenfunctions associated with the (modified) Kadomtsev-Petviashvili (K P) hierarchy form generators of a symmetry transformation. Linear inte gro-differential representations for these symmetries are found. For s pecial cases the corresponding nonlinear equations are the compatibili ty conditions of linear scattering problems of Loewner type. The examp les include the 2+1-dimensional sine-Gordon equation with space variab les occuring on an equal footing introduced recently by Konopelchenko and Rogers. This equation represents a special squared eigenfunction s ymmetry of the Ishimori hierarchy.