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In the late 1980s, Premet conjectured that the nilpotent variety of anyfinite dimensional restricted Lie algebra over an algebraically closed field ofcharacteristic $p>0$ is irreducible. This conjecture remains open, but it isknown to hold for a large class of simple restricted Lie algebras, e.g. for Liealgebras of connected reductive algebraic groups, and for Cartan series $W, S$and $H$. In this paper, with the assumption that $p>3$, we confirm thisconjecture for the minimal $p$-envelope $W(1;n)_p$ of the Zassenhaus algebra$W(1;n)$ for all $n\geq 2$.

The nilpotent variety of $W(1;n)_{p}$ is irreducible