**Název anglicky / Title English:**Integrability versus separability in a magnetic field

**Osnova / Outline:**

It is known that in the absence of magnetic field, integrable Hamiltonian systems are separable if and only if a pair of quadratic commuting integrals exists. When the system possesses only higher order integrals even superintegrability does not force separability. In presence of magnetic fields the relationship between separability and integrability is lost already with quadratic integrals.

Thus, a question arises on what to do with such systems that are integrable, but not separable, i.e. systems for which the ability to integrate the equations of motion does not guarantee the explicit construction of the integrating variables (in position space).

The task here is

1) to study and understand the basic definitions related to the topic, i. e. the concepts of integrability, superintegrability, separability and their interrelationship,

2) to get acquainted with known results on three dimensional integrable and superintegrable systems in a magnetic field,

3 ) to launch its own research in the field of (super)integrable but not separable systems with a magnetic field.

**Literatura / reference:**

1) Eisenhart, L. P., Separable systems of Stackel. Ann. of Math. (2) 35 (1934), no. 2, 284–305.

2) Miller, W., Jr.; Post, S.; Winternitz, P., Classical and quantum superintegrability with applications. J. Phys. A 46 (2013), no. 42, 423001, 97 pp.

3) Charest, F.; Hudon, C.; Winternitz, P., Quasiseparation of variables in the Schrödinger equation with a magnetic field. J. Math. Phys. 48 (2007), no. 1, 012105, 16 pp.

4) Kubů, O.; Marchesiello, A., Šnobl, L. Superintegrability of separable systems with magnetic field: the cylindrical case. J. Phys. A 54 (2021), no. 42, Paper No. 425204, 38 pp.