  Rota-Baxter systems of T. Brzezi\'{n}ski are a generalization of Rota-Baxter
operators that are related to dendriform structures, associative Yang-Baxter
pairs and covariant bialgebras. In this paper, we consider Rota-Baxter systems
in the presence of bimodule, which we call generalized Rota-Baxter systems. We
define a graded Lie algebra whose Maurer-Cartan elements are generalized
Rota-Baxter systems. This allows us to define a cohomology theory for a
generalized Rota-Baxter system. Formal one-parameter deformations of
generalized Rota-Baxter systems are discussed from cohomological points of
view. We further study Rota-Baxter systems, associative Yang-Baxter pairs,
covariant bialgebras and introduce generalized averaging systems that are
related to associative dialgebras. Next, we define generalized Rota-Baxter
systems in the homotopy context and find relations with homotopy dendriform
algebras. The paper ends by considering commuting Rota-Baxter systems and their
relation with quadri-algebras.
