  A generalisation of the notion of a Rota-Baxter operator is proposed. This
generalisation consists of two operators acting on an associative algebra and
satisfying equations similar to the Rota-Baxter equation. Rota-Baxter operators
of any weights and twisted Rota-Baxter operators are solutions of the proposed
system. It is shown that dendriform algebra structures of a particular kind are
equivalent to Rota-Baxter systems. It is shown further that a Rota-Baxter
system induces a weak peudotwistor [F. Panaite & F. Van Oystaeyen, Twisted
algebras, twisted bialgebras and Rota-Baxter operators, arXiv:1502.05327
(2015)] which can be held responsible for the existence of a new associative
product on the underlying algebra. Examples of solutions of Rota-Baxter systems
are obtained from quasitriangular covariant bialgebras hereby introduced as a
natural extension of infinitesimal bialgebras [M. Aguiar, Infinitesimal Hopf
algebras, [in:] New trends in Hopf algebra theory (La Falda, 1999), Contemp.
Math., 267, Amer. Math. Soc., Providence, RI, (2000), pp. 1-29].
