On the Existence of the Conditionally Linear Integral in Conservative Holonomic Systems with Two Degrees of Freedom

For the Lagrange equations of the 2nd kind the problem of the existence of the conditionally linear in the velocities integral, which possesses the property that its total time derivative is identically proportional to the integral itself, is considered. The Lagrange function is assumed to be given in arbitrary generalized coordinates. The conditions for the existence of such integral are reduced to the study of the compatibility of two equations in partial derivatives of the 2nd order for one unknown function of two independent arguments. These equations are written in the invariant form in an arbitrary system of generalized coordinates and the problem is transformed into the investigation of the set of Pfaffian equations.