Pairs of commuting quadratic elements in the universal enveloping algebra of Euclidean algebra and integrals of motion* *Inspiration and need for this paper came from numerous discussions with Pavel Winternitz over several years. Unfortunately, while he was alive we always found other more urgent research directions to follow together. Thus we can only dedicate our paper to his memory.

Abstract

Motivated by the consideration of integrable systems in three spatial dimensions in Euclidean space with integrals quadratic in the momenta we classify three-dimensional Abelian subalgebras of quadratic elements in the universal enveloping algebra of the Euclidean algebra under the assumption that the Casimir invariant vanishes in the relevant representation. We show by means of an explicit example that in the presence of magnetic field, i.e. terms linear in the momenta in the Hamiltonian, this classification allows for pairs of commuting integrals whose leading order terms cannot be written in the famous classical form of Makarov et al [17]. We consider limits simplifying the structure of the magnetic field in this example and corresponding reductions of integrals, demonstrating that singularities in the integrals may arise, forcing structural changes of the leading order terms.