  This article is devoted to completing some aspects of the classical
Cauchy-Lipschitz (or Picard-Lindel\"of) theory for general nonlinear systems
posed on time scales, that are closed subsets of the set of real numbers.
Partial results do exist but do not cover the framework of general dynamics on
time-scales encountered e.g. in applications to control theory. In the present
work, we first introduce the notion of absolutely continuous solution for
shifted and non shifted Delta-Cauchy problems, and then the notion of a maximal
solution. We state and prove a Cauchy-Lipschitz theorem, providing existence
and uniqueness of the maximal solution of a given Delta-Cauchy problem under
suitable assumptions like regressivity and local Lipschitz continuity, and
discuss some related issues like the behavior of maximal solutions at terminal
points.
