Superintegrable metrics on surfaces admitting integrals of degrees 1 and 4

We study Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral quartic in momenta. The main results of the work are local description of such metrics in terms of ordinary differential equations, integration of the equations, and description of the corresponding Poisson algebra of integrals of motion. We also give examples of such metrics that can be extended to the sphere $S^2$, and study the group of symmetries of the problem.