Superintegrability of Geodesic Motion on the Sausage Model

Understand

Reduction of the $\eta$-deformed sigma model on ${\rm AdS}_5 \times {\rm S}^5$ to the two-dimensional squashed sphere $({\rm S}^2)_{\eta}$ can be viewed as a special case of the Fateev sausage model where the coupling constant $\nu$ is imaginary.

• We show that geodesic motion in this model is described by a certain superintegrable mechanical system with four-dimensional phase space.
• This is done by means of explicitly constructing three integrals of motion which satisfy the $\mathfrak{sl}(2)$ Poisson algebra relations, albeit being non-polynomial in momenta.
• Further, we find a canonical transformation which transforms the Hamiltonian of this mechanical system to the one describing the geodesic motion on the usual two-sphere.