HyperKhaler Metrics Building and Integrable Models

Abstract

Methods developed for the analysis of integrable systems are used to study the problem of hyperK\"ahler metrics building as formulated in D=2 N=4 supersymmetric harmonic superspace. We show, in particular, that the constraint equation $\beta\partial^{++2}\omega -\xi^{++2}\exp 2\beta\omega =0$ and its Toda like generalizations are integrable. Explicit solutions together with the conserved currents generating the symmetry responsible of the integrability of these equations are given. Other features are also discussed