Algebraic roots of integrability of nonlinear evolution equations and dressing procedure
Commutator identities on associative algebras generate solutions of linearized versions of integrable equations. We realize elements of associative algebra in a class of integral operators kernels of which are belonging to the space of distributions. In this class we impose some generic conditions on the representation and develop a dressing procedure that enables to derive both: nonlinear integrable equation itself and its Lax pair associated to the given commutator identity. Thus problem of construction of new
integrable nonlinear evolution equations is reduced to the problem of construction of commutator identities on associative algebras. Different examples of this construction that involves (1+1) and (2+1) nonlinear
differential equations, as well as difference-differential equations, are considered.