Conjecture on self-annihilating invariance vector fields for Lie–Poisson systems

Let the space $\mathcal{P}$ coincide with the dual space of a Lie algebra $\mathfrak{g}$, and let the Poisson brackets $\{\,\ \}_1$ coincide with the standard Lie–Poisson brackets on $\mathfrak{g}^*$. An invariance vector field is a vector field satisfying

$$Z(q_i)=Z(p_i)=0,\qquad i\in\{1,\dots,n\}.$$

Self-annihilating invariance-vector-field conjecture. Among the invariance vector fields there exists a special vector field $Z$ which, acting in the system of natural Lie-algebraic coordinates, namely the linear coordinate functions on $\mathfrak{g}^*$, annihilates its own components.

This conjecture concerns the existence of a particularly normalized invariance vector field in the Lie–Poisson setting. The supplied text gives no evidence of a proof or disproof, so its resolution remains open.