  In general, quantum matrix algebras are associated with a couple of
compatible braidings. A particular example of such an algebra is the so-called
Reflection Equation algebra. In this paper we analyse its specific properties,
which distinguish it from other quantum matrix algebras (in first turn, from
the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity
for its generating matrix, which in a limit turns into the Cayley-Hamilton
identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we
consider some specific properties of the braided Yangians, recently introduced
by the authors. In particular, we establish an analog of the Cayley-Hamilton
identity for the generating matrix of such a braided Yangian. Besides, by
passing to a limit of the braided Yangian, we get a Lie algebra similar to that
entering the construction of the rational Gaudin model. In its enveloping
algebra we construct a Bethe subalgebra by the method due to D.Talalaev.
