New integrability properties of a family of sequences of ordinary differential equations, which contains the Riccati and Abel chains as the most simple sequences, are studied. The determination of n generalized symmetries of the nth-order equation in each chain provides, without any kind of integration, n-1 functionally independent first integrals of the equation. A remaining first integral arises by a quadrature by using a Jacobi last multiplier that is expressed in terms of the preceding equation in the corresponding sequence. The complete set of n first integrals is used to obtain the exact general solution of the nth-order equation of each sequence. The results are applied to derive directly the exact general solution of any equation in the Riccati and Abel chains.

• 34A05 - Explicit solutions, first integrals of ordinary differential equations
• 34A33 - Ordinary lattice differential equations
• 34C14 - Symmetries, invariants of ordinary differential equations
• 37J35 - Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests