Vladimir Sergeevich Matveev

We construct Lax pairs for the recently (2023) introduced integrable PDE systems known as the BKM equations. As many known and previously studied integrable systems are special cases of the BKM systems, our construction provides Lax pairs for many integrable hierarchies, including previously studied ones such as Camassa–Holm, Dullin–Gottwald–Holm, cKdV, Ito, and Marvan–Pavlov, as well as new ones. The corresponding pair is related to a Sturm–Liouville operator on the real line whose potential depends rationally on the spectral parameter.

We study existence and nonexistence of diagonal and separating coordinates for Riemannian symmetric spaces of rank 1. We generalize the results of Gauduchon and Moroianu (2020) by showing that a symmetric space of rank 1 has diagonal coordinates if and only if it has constant sectional curvature. This implies that orthogonal separation of variables on a symmetric space of rank 1 is possible only in the constant sectional curvature case. We show that on the complex projective space ℂ P n and on complex hyperbolic space ℂ H n , with n ≥ 2 , separating coordinates necessarily have precisely n ignorable coordinates. In view of results of Boyer et al. (1983, 1985) and later results of Winternitz et al. (1994), this completes the description of separation of variables on ℂ P n for all n and on ℂ H n for n = 2 , 3 .

We show that for a hyperbolic system of hydrodynamic type admitting n symmetries, there exists a coordinate system in which the generator of the system and all the symmetries are diagonal.

We show that if 𝑛 functionally independent commutative quadratic in momenta integrals for the geodesic flow of a Riemannian or pseudo-Riemannian metric on an 𝑛-dimensional manifold are simultaneously diagonalisable at the tangent space to every point, then they come from the Stäckel construction, so the metric admits orthogonal separation of variables.

We construct Lax pairs for the recently (2023) introduced integrable PDE systems known as the BKM equations. As many known and previously studied integrable systems are special cases of the BKM systems, our construction provides Lax pairs for many integrable hierarchies, including previously studied ones such as Camassa-Holm, Dullin-Gottwald-Holm, cKdV, Ito, and Marvan-Pavlov, as well as new ones. The corresponding pair is related to a Sturm-Liouville operator on the real line whose potential depends rationally on the spectral parameter.

In this paper, we prove a recent conjecture of [V. Dragović, B. Gajić and B. Jovanović, Integrability of homogeneous exact magnetic flows on spheres, Regul. Chaotic Dyn. 30(4) (2025) 582–597, https://doi.org/10.1134/S1560354725040082 ] stating that the magnetic geodesic flow on the standard sphere [ ] whose magnetic 2-form is the restriction of a constant 2-form from [ ] is Liouville integrable. The integrals are quadratic and linear in momenta.

Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler–Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.

We show that quadratic Hamiltonians in involution coming from a Stäckel system are quantizable, in the sense that one can construct commutative self-adjoint operators whose symbols are the quadratic Hamiltonians. Moreover, they allow multiplicative separation of variables. This proves a conjecture explicitly formulated in Błaszak et al (2016 Ann. Phys., NY 371 460–77).