  This paper explores a relationship between invariants of certain group
actions and the time-reversibility of two-dimensional polynomial differential
systems exhibiting a $1:-1$ resonant singularity at the origin. We focus on the
connection of time-reversibility with the Sibirsky subvariety of the center
(integrability) variety, which encompasses systems possessing a local analytic
first integral near the origin. An algorithm for generating the Sibirsky ideal
for these systems is proposed and the algebraic properties of the ideal are
examined.
  Furthermore, using a generalization of the concept of time-reversibility we
study $n$-dimensional systems with a $1:\zeta:\zeta^2:\dots:\zeta^{n-1}$
resonant singularity at the origin, where $n$ is prime and $\zeta$ is a
primitive $n$-th root of unity. We study the invariants of a Lie group action
on the parameter space of the system, leveraging the theory of binomial ideals
as a fundamental tool for the analysis. Our study reveals intriguing
connections between generalized reversibility, invariants, and binomial ideals,
shedding light on their complex interrelations.
