Integro-differential non-linear equations and continual Lie algebras

The integrability problem of integro-differential equations with, generally speaking, singular kernels is discussed after an example of new continual analogs of the two-dimensional Toda lattices. These equations are associated with new infinite-dimensional Lie algebras via zero curvature type representation. The structural constants of these algebras are distributions. A formal solution of the Goursát problem is obtained. For the case with the kernel of the integral operator being _±-distribution an explicit expression in quadratures for the solutions is given.