q-cut-and-join operators related to Reflection Equation algebras

The classical cut-and-joint operators, playing a very important role in Hurwitz theory, differ from the Casimir operators, coming from the enveloping algebras U(gl(N)), by normal ordering. Construction of the cut-and-join operators is based on the fact that the product of two square matrices L=MD, where M has commutative entries and D is composed from the partial derivatives in these entries, generates the algebra U(gl(N)). I'll define analogs of all these objects related to the so-called Reflection Equation algebras and describe a way to perform spectral analysis of the corresponding operators. Also, I plan to exhibit other applications of the formula L=MD in the "q-setting".