  As a generalization of skew braces, the notion of skew trusses was introduced
by T. Brzezinski. It was shown that every Rota-Baxter group has the structure
of skew braces by V. G. Bardakov and V. Gubarev. To investigate an analogue of
Rota-Baxter groups which has the structure of skew trusses, we define
RotaBaxter systems. We study the relationship between Rota-Baxter systems and
Rota-Baxter groups. Furthermore, we prove that a Rota-Baxter system can be
decomposed as a direct sum of two semigroups. A factorization theorem is
proved, generalizing the factorization theorem of Rota-Baxter groups. The
notion of Rota-Baxter systems of Lie algebras was introduced, as a
generalization of Rota-Baxter Lie algebras. The connection between Rota-Baxter
systems of Lie algebras and Lie groups is studied. Finally, as a generalization
of the modified Yang-Baxter equation, we define twisted modified Yang-Baxter
equations. We give solutions of twisted modified Yang-Baxter equations by
Rota-Baxter systems of Lie algebras.
