We first investigate the interconnection of invariants of certain group actions and time-reversibility of a class of two-dimensional polynomial systems with 1:-1 resonant singularity at the origin. 

The time-reversibility is related to the Sibirsky subvariety of the center (integrability) variety and it is known that every time-reversible system has a local analytic first integral at the origin. 

We propose a new  algorithm to obtain a generating set for the Sibirsky ideal of such polynomial  systems and investigate  some algebraic properties of this ideal.

Then, we discuss a generalization of the concept of time-reversibility in the three dimensional case considering the systems with 1:z:z^2 resonant singularity at the origin (where z is a primitive cubic root of unity) and study a connection of such reversibility  with the invariants of some group actions in the space of parameters of the system and Lawrence ideals.

This is a joint work with Mateja Grasic.