We consider the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetime without imposing any restrictions on the functional parameters characterizing the metric. We establish commutativity of the second-order operators constructed from the Killing tensors found in [J. High Energy Phys. 02 (2007) 004] and show that these operators, along with the first-order operators originating from the Killing vectors, form a complete set of commuting symmetry operators (i.e., integrals of motion) for the Klein-Gordon equation. Moreover, we demonstrate that the separated solutions of the Klein-Gordon equation obtained in [J. High Energy Phys. 02 (2007) 005] are joint eigenfunctions for all of these operators. We also present explicit form of the zero mode for the Klein-Gordon equation with zero mass.In the semiclassical approximation we find that the separated solutions of the Hamilton-Jacobi equation for geodesic motion are also solutions for a set of Hamilton-Jacobi-type equations which correspond to the quadratic conserved quantities arising from the above Killing tensors.

The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface elevation for a prescribed relation between the orders of the two expansion parameters, the amplitude parameter α and the long wavelength (or shallowness) parameter β, is developed. Unlike the heuristic approaches found in the literature, when modifications are made in the equation for surface elevation itself, the procedure starts from the consistently truncated asymptotic expansions for unidirectional waves, a counterpart of the Boussinesq system of equations for the surface elevation and the bottom velocity, from which the leading order and higher order equations for the surface elevation can be obtained by iterations. The relations between the orders of the two small parameters are taken in the form β = O(α n ) and α = O(β m ) with n and m specified to some important particular cases. The analysis shows, in particular, that some evolution equations, proposed before as model equations in other physical contexts (like the Gardner equation, the modified KdV equation, and the so-called 5th-order KdV equation), can emerge as the leading order equations in the asymptotic expansion for the unidirectional water waves, on equal footing with the KdV equation. The results related to the higher orders of approximation provide a set of consistent higher order model equations for unidirectional water waves which replace the KdV equation with higher-order corrections in the case of non-standard ordering when the parameters α and β are not of the same order of magnitude. The shortcomings of certain models used in the literature become apparent as a result of the subsequent analysis. It is also shown that various model equations obtained by assuming a prescribed relation β = O(α n ) between the orders of the two small parameters can be equivalently treated as obtained by applying transformations of variables which scale out the parameter β in favor of α. It allows us to consider the nonlinearity-dispersion balance, epitomized by the soliton equations, as existing for any β, provided that α → 0, but leads to a prescription, in asymptotic terms, of the region of time and space where the equations are valid and so the corresponding dynamics is expected to occur.

We present a multiparameter generalization of the Stäckel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stäckel transform preserves Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation.Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized Stäckel transform.

We present a simple novel construction of recursion operators for integrable multidimensional dispersionless systems that admit a Lax pair whose operators are linear in the spectral parameter and do not involve the derivatives with respect to the latter. New examples of recursion operators obtained using our technique include inter alia those for the general heavenly equation, which describes a class of anti-selfdual solutions of the vacuum Einstein equations, and a six-dimensional equation resulting from a system of Ferapontov and Khusnutdinova.

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all known today recursion operators and are much easier to verify than those found in earlier work.We also give explicit formulas for the nonlocal parts of higher recursion, Hamiltonian and symplectic operators of integrable systems in (1+1) dimensions.Using these two results we prove, under some natural assumptions, the Maltsev-Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.