A (2+1)-dimensional quasilinear system is said to be 'integrable' if it can be decoupled in infinitely many ways into a pair of compatible n-component onedimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogs of n-gap solutions. It is demonstrated that the requirement of the existence of 'sufficiently many' n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.MSC: 35L40, 35L65, 37K10.

We obtain the necessary and sufficient conditions for a two-component (2+1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in involution, implying that the systems in question are locally parametrized by 15 arbitrary constants. It is proved that all such systems possess three conservation laws of hydrodynamic type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such systems are proved to possess a scalar pseudopotential which plays the role of the 'dispersionless Lax pair'. We demonstrate that the class of two-component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions, thus establishing the equivalence of the two possible definitions of the integrability. Explicit linearly degenerate examples are constructed.MSC: 35L40, 35L65, 37K10.

A (d + 1)-dimensional dispersionless PDE is said to be integrable if its ncomponent hydrodynamic reductions are locally parametrized by (d − 1)n arbitrary functions of one variable. Given a PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions (if consistent).MSC: 35L40, 35L65, 37K10.

We investigate integrable second order equations of the form F (u xx , u xy , u yy , u xt , u yt , u tt ) = 0.Familiar examples include the Boyer-Finley equation u xx + u yy = e utt , the potential form of the dispersionless Kadomtsev-Petviashvili (dKP) equation u xt − 1 2 u 2 xx = u yy , the dispersionless Hirota equation (α − β)e uxy + (β − γ)e uyt + (γ − α)e utx = 0, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit. MSC: 35Q58, 37K25, 53A40, 53B25, 53Z05.