Semi-invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector-eigenvalue pair for linear systems, and are, therefore, very useful to depict the behaviour of the system. In this article, semi-invariants are used to construct explicitly Lyapunov functions useful for studying the stability of the origin, for continuous-time systems. Moreover, using well-known tools from differential geometry such as (orbital) symmetries, it is shown how semi-invariants can be found for several classes of systems. Important connections with centre manifold theory are pointed out. By using the proposed general techniques, a new proof of a result claimed to Bendixson (Bendixson, I. (1901), 'Sur les courbes definies par des equations differentielles', Acta Mathematics, 24, 1-88) is given.