This paper contains a general description of the theory of invariants under the adjoint action of a given finite‐dimensional complex Lie algebra G, with special emphasis on polynomial and rational invariants. The familiar ’’Casimir’’ invariants are identified with the polynomial invariants in the enveloping algebra U (G). More general structures (quotient fields) are required in order to investigate rational invariants. Some useful criteria for G having only polynomial or rational invariants are given. Moreover, in most of the physically relevant Lie algebras the exact computation of the maximal number of algebraically independent invariants turns out to be very easy. It reduces to finding the rank of a finite matrix. We apply the general method to some typical examples.

A general scheme for analysing reductions of dispersionless integrable hierarchies is presented. It is based on a method for determining the S-function by means of a system of first-order differential equations. Compatibility systems of nonlinear partial differential equations of Bourlet type characterizing both reductions and hodograph solutions of the dKP hierarchy are obtained. Wide classes of illustrative explicit examples are exhibited.

The quasi-classical limit of the scalar nonlocal dbar-problem is derived and a quasi-classical version of the dbar-dressing method is presented. Dispersionless KP, mKP and 2DTL hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy it is nothing but the ring of symmetries for the quasi-classical dbar-problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.Comment: LaTex file,19 page