  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.
