A geometric approach to higher-order Riccati chain: Darboux polynomials and constants of the motion

The properties of higher-order Riccati equations are investigated. The second-order equation is a Lagrangian system and can be studied by using the symplectic formalism. The second-, third- and fourth-order cases are studied by proving the existence of Darboux functions. The corresponding cofactors are obtained and some related properties are discussed. The existence of generators of t-dependent constants of motion is also proved and then the expressions of the associated time-dependent first integrals are explicitly obtained. The connection of these time-dependent first integrals with the so-called master symmetries, characterizing some particular Hamiltonian systems, is also discussed. Finally the general n-th-order case is analyzed.