Note on the Functional Equations

which is a nonalgebraic integral of the system (1; 1, 0). This conclusion contradicts the fact of Fo being an algebraic function, therefore the initial hypothesis that assumes F to be dependent upon x3 and y3 is absurd. F being independent of x3 and y3, it follows, as initially said, that any algebraic integral of the system (1; 1, 1) is an algebraic combination of the four integrals (12; 1,1), that is, such a system does not admit any other algebraic integral distinct from the integrals (12; 1,1).