  We study integrable systems on the semidirect product of a Lie group and its
Lie algebra as the representation space of the adjoint action. Regarding the
tangent bundle of a Lie group as phase space endowed with this semidirect
product Lie group structure, we construct a class of symplectic submanifolds
equipped with a Dirac bracket on which integrable systems (in the
Adler-Kostant-Symes sense) are naturally built through collective dynamics. In
doing so, we address other issues as factorization, Poisson-Lie structures and
dressing actions. We show that the procedure becomes recursive for some
particular Hamilton functions, giving rise to a tower of nested integrable
systems.
