  In this paper, we propose the concept of an $\Omega$-Rota-Baxter system,
which is a generalization of a Rota-Baxter system and an $\Omega$-Rota-Baxter
algebra of weight zero. In the framework of operated algebras, we obtain a
linear basis of a free $\Omega$-Rota-Baxter system for an extended
diassociative semigroup $\Omega$, in terms of bracketed words and the method of
Gr\"obner-Shirshov bases. As applications, we introduce the concepts of
Rota-Baxter system family algebras and matching Rota-Baxter systems as special
cases of $\Omega$-Rota-Baxter systems, and construct their free objects.
Meanwhile, free $\Omega$-Rota-Baxter algebras of weight zero, free Rota-Baxter
systems, free Rota-Baxter family algebras and free matching Rota-Baxter
algebras are reconstructed via new method.
