Three-dimensional dynamical systems admitting nonlinear superposition with three-dimensional Vessiot-Guldberg-Lie algebras (Q901009)

The well-known Bianchi classification of non-isomorphic structures of \(3\)-dimensional real Lie algebras is used in this paper in order to classify \(3\)-dimensional dynamical systems admitting nonlinear superpositions into thirty one standard types presented in Tables 2 and 3. Twenty four of them, presented in Table 2, are associated with solvable Vessiot-Guldberg-Lie algebras and can be reduced to systems of first-order linear equations. The remaining seven standard types, presented in Table 3, are nonlinear and hence their integration remains an open problem.