## ABSTRACT

The term “separation of variables” (SoV) appears in various branches of mathematical sciences, due to the preponderance of diﬀerential equations. In a sense, it constitutes the simplest approach toward attempting a solution of any dynamical problem. Starting from the simplest differential equations to those governing complicated natural phenomena, its utility has been repeatedly proved. Contrary to the usual notions, the system of equations may be either a set of partial diﬀerential equations or ordinary diﬀerential equations. In the former, the term SoV usually refers to the separation of the actual coordinates, which may be cartesian or polar variables, as in the hydrogen atom. In the latter, the term refers to the separation of independent dynamical variables. In a completely integrable system, separation of variables assumes utmost importance, as such systems are endowed with an inﬁnite number of conserved quantities. The notion of complete integrability is entwined with the concept of separability. Classical studies of dynamical systems focused on identifying “action-angle” variables, thus in essence achieving separability. Furthermore, it is well known that integrable systems are often bi-or even multi-Hamiltonian in character, and in such cases separation of variables has often been an additional output of this characteristic feature.