Existence of a Center Manifold in a Practical Domain around L1 in the Restricted Three Body Problem

We give a proof of existence of centre manifolds within large domains for systems with an integral of motion. The proof is based on a combination of topological tools, normal forms and rigorous-computer-assisted computations. We apply our method to obtain an explicit region in which we prove existence of a center manifold in the planar Restricted Three Body Problem.