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

Superintegrable metrics on surfaces admitting integrals of degrees 1 and 4