We consider two-dimensional relativistically invariant systems with a three-dimensional reducible configuration space and a chiral-type Lagrangian that admit higher symmetries given by polynomials in derivatives up to the fifth order. Nine such systems are known: two are Liouville-type systems, and zero-curvature representations for two others have previously been found. We here give zero-curvature representations for the remaining five systems. We show how infinite series of conservation laws can be derived from the established zero-curvature representations. We give the simplest higher symmetries; others can be constructed from the conserved densities using the Hamiltonian operator. We find scalar formulations of the spectral problems.
A three-dimensional integrable generalization of the Stäckel systems is proposed. The classification of such systems is obtained which results in two families. The first one is the direct sum of the two-dimensional case which is equivalent to the representation of the Schottky-Manakov top in the quasi-Stäckel form and a Stäckel one-dimensional system. The second family is probably a new three-dimensional system. The system of hydrodynamic type, which we get from this family in a usual way, is a 3-dimensional generalization of the Gibbons-Tsarev system. A generalization of the quasi-Stäckel systems to the case of any dimension is discussed.
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