Scattering transformation at fixed non-zero energy for the two-dimensional Schrodinger operator with potential decaying at infinity

We study the problem of reconstructing the potential of the two-dimensional Schrodinger operator from scattering data measured at fixed energy. This problem. in contrast to the general multidimensional inverse problem, possesses an infinite-dimensional symmetry algebra generated by the Novikov-Veselov hierarchy and hence is 'exactly soluble' in some sense; the complexity of the answer is approximately the same as in the one-dimensional problem. We make heavy use of methods developed in modern soliton theory. Since the quantum fixed-energy scattering problem is mathematically equivalent to the acoustic single-frequency scattering problem? we sec that the results of the present paper apply in both cases.