The integer and linear constraint satisfaction problem, which consists in proving the emptiness of the set of integer points satisfying a set of linear constraints or the existence of a solution, is very frequent in the field of computer science (vectorization, code scheduling, etc.). Most methods proposed in the literature deal with various specific instances of this problem. In this paper, the problem is considered in its general form. Some standard methods are analyzed. Some new algorithms are proposed, either to simplify the problem, or to solve exactly (by cutting plane methods) the reduced for of the problem. A sequence of such algorithms is implemented in the automatic parallelizer available under PIAF, an interactive programming environment for FORTRAN. However, these algorithms could be applied to more difficult problems than the usual ones appearing in data dependence analysis.