Compatibility conditions for a non-quadratic integral of motion

Conditions are found which are satisfied by the coefficients of the expression being a second integral of the motion of an autonomous dynamical system with two degrees of freedom. The coefficientsA, B. , ,E are differentiable functions of the cartesian position coordinatesx, y. The velocity components are denoted by . It is shown that must be constant andB must be of the formB =f(x+y) +g(x-y) wheref, g are arbitrary.Given andB one can always find the remaining coefficientsA, E and also the corresponding potential and second integral. Depending on the specifica case at hand a certain number of arbitrary constants (or arbitrary functions) enter into the potential and the second integral. To each potential (which may be of the separable or nonseparable type in the coordinatesx andy)there corresponds one integral of the above form.