A heuristic scheme is described for constructing Lyapunov v-functions, generalizing the classical method for constructing these functions from the first integrals of the equations of motion under investigation (or from the integrals of a comparison system). It is shown that the generalized scheme inherits a characteristic feature of the classical method: the Lyapunov functions are constructed as solutions of a certain completely integrable partial differential equation (or system of such equations). The form of this equation and its order are uniquely defined by a non-degenerate multi-parameter function V(x, a) + %, x ~ R n, a E R 'rq (where a is a parameter vector), which generalizes the classical linear combination of integrals. Methods are described for representing v-functions, in the course of which the traditional methods (the method of Chetayev combinations of integrals and the construction of Lyapunov functions as a non-linear function of integrals) are augmented by geometrical constructions in which the v-functions are sought in the form of envelopes of certain subfamilies of the function V(x, a) + %. The generalized scheme serves as a basis for deriving new, simple criteria for the asymptotic stability of the trivial solution in a transcendental problem of the stability of a system with two degrees of freedom in the critical case of two pairs of pure imaginary roots at 1 : 1 resonance (the case of simple elementary divisors).