Time-reversibility and ivariants of some $3$-dim systems

We study time-reversibility and invariants of the group of transformations $x\to x, ~y\to \alpha y, ~z \to \alpha ^{-1}z$ for three-dimensional polynomial systems with $0:1:-1$ resonant singular point at the origin. An algorithm to find the Zariski closure of the set of time-reversible systems in the space of parameters is proposed. The interconnection of time-reversibility and invariants of the group mentioned above is discussed.