A special case of integrability in a system with one quasi-cyclic coordinate

It is well known that conservative holonomic and scleromic systems with two degrees of freedom which have one cyclic coordinate are ‘integrable’. This means that the solution to the equations of motion can be given analytically in terms of quadratures, due to the existence of the two first integrals: the energy integral and the integral corresponding to the cyclic coordinate. In the present paper it is shown that the system is ‘integrable’ even if it is only holonomic and scleronomic and has one ‘quasi-cyclic’ coordinate, and even if the generalized forces are non-conservative provided the kinetic energy satisfies a certain additional condition.