Rational integrals of the second degree of two-dimentional geodesic equations

For projection of two-dimensional geodesic equations we consider the problem of finding integrals that are rational in generalized velocities. We obtain the conditions of the existence of integral in the form of the quotient of polynomials of the second degree when the denominator is a squared linear polynomial. In general case first condition of the existence of the rational integral of the second degree is given. Integrals in the form of the quotient of polynomials of the first, second, third and fourth degree are constructed for the simplest case of symmetric metrics.