The basic object of investigation is classical Chazy - a nonlinear
third-order autonomous differential equation with movable singular line.
In this paper, the first integral of Chazy equation is obtained with the
help of Backlund transformation, which is a transcendental dependence on
the function and its derivatives . The general solution of equation is
represented by Laurent series in the punctured neighborhood of point at
infinity, and in the domain is represented by Taylor series. It is
proved that the obtained Laurent and Taylor series all converge
absolutely in their defined domains. Under the situation that the
analytic continuation of solutions of differential equation with movable
singular line cannot be realized, a necessary and sufficient condition
is presented for obtaining the two-parameter rational solution of Chazy
equation from its general solution constructed by Taylor series. And the
research results can be applied in the analytic theory of ordinary
differential equations.