  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.
