Abstract
In autonomous dynamical systems with two degrees of freedom, any integral polynomial in the velocities determines uniquely a complex analytic function. Conversely, given any analytic function there exists a polynomial integral of a dynamical system which determines this function. The analytic function provides an immediately verifiable necessary criterion which can answer the questions of whether an expression which is polynomial in the velocities can be an integral of any (unknown) two-dimensional dynamical system. The basic operation among integrals corresponds to similar operations among the corresponding analytic functions.
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