By using the self-dual Yang-Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by non-auto-Backlund transformation. We prove Lie-algebra isomorphism between SDYM those potential (PSDYM) equation, describe construction these systems. Using certain known aspects PSDYM symmetry algebra...

We write down the Poisson structure for a relativistic particle where the Lorentz group does not act canonically, but instead as a Poisson-Lie group. In so doing we obtain the classical limit of a particle moving on a noncommutative space possessing SL q (2, C) invariance. We show that if the standard mass shell constraint is chosen for the Hamiltonian function, then the particle interacts with...

Reduced Ermakov systems are defined as Ermakov systems restricted to the level surfaces of the Ermakov invariant. The condition for Lie point symmetries for reduced Ermakov systems is solved yielding four infinite families of systems. It is shown that SL(2, R) always is a group of point symmetries for the reduced Ermakov systems. The theory is applied to a model example and to the equations of ...

In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Wacławczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extend...