Classical and quantum integrability in dimensions two and three

The thesis consists of two parts. In the first part, we study the integrability of geodesic flows on 3-manifolds that admit $\widetilde{SL(2,\mathbb R)}$ geometry in Thurston's sense. The main examples are the quotients $\mathcal M^3_\Gamma=\Gamma\backslash PSL(2,\mathbb R)$, where $\Gamma \subset PSL(2,\mathbb R)$ is a cofinite Fuchsian group. We show that the corresponding phase space $T^*\mathcal M_\Gamma^3$ contains two open regions with integrable and chaotic behaviour. In the integrable region we have Liouville integrability with analytic integrals, while in the chaotic region the system is not Liouville integrable even in smooth category and has positive topological entropy.

In the concrete example of the modular group $\Gamma=PSL({2,\mathbb Z})$ we extend the link of periodic geodesics with knot theory, discovered by Ghys, to the integrable region.

In the second part, we consider the integrable generalisations of the Dirac magnetic monopole on sphere $\mathbb S^2$ with general metric and constant non-zero density of magnetic field. We complete the local classification results of such systems by Ferapontov, Sayles and Veselov, extending them to the analytic integrable systems on the topological sphere. In the limiting even case we have the new integrable two-centre problem on the usual round sphere in the external field of Dirac magnetic monopole.