Universal enveloping Lie Rota-Baxter algebra of preLie and post-Lie algebras

Universal enveloping Lie Rota-Baxter algebras of pre-Lie and post-Lie algebras are constructed. It is proved that the pairs of varieties (Lie Rota-Baxter algebras of zero weight,preLie algebras) and (Lie Rota-Baxter algebras of nonzero weight,post-Lie algebras) are PBW-pairs and the variety of Lie Rota-Baxter algebras is not Schreier.