N ov 2 00 6 A hierarchy of integrable PDEs in 2 + 1 dimensions associated with 2 - dimensional vector fields

We introduce a hierarchy of integrable partial differential equations in 2 + 1 dimensions arising from the commutation of 2 dimensional vector fields, and we construct the formal solution of the associated Cauchy problems using the Inverse Scattering Transform for one-parameter families of vector fields recently introduced in [1, 2]. Due to the ring property of the space of eigenfunctions, the inverse problem can be formulated in three distinguished ways; in particular, one formulation corresponds to a linear integral equation for a Jost eigenfunction, and another formulation is a scalar nonlinear Riemann problem for suitable analytic eigenfunctions.