Let $L$ be a simple Lie algebra of Cartan type of absolute toral rank 2 over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq 5$. Denote by $L_{[p]} $ the minimal $p$-envelope of $L$ and let ${\mathfrak t}$ be a 2-dimensional torus of $L_{[p]}$. For a nontrivial irreducible restricted $L_{[p]}$-module $V$ we denote by $\Gamma(V, {\mathfrak t})$ the set of weights of $V$ with respect to ${\mathfrak t}$.

In Section 10.7 of the book [H. Strade: Simple Lie algebras over fields of positive characteristic. II. Classifying the absolute toral rank two case] it is showed that $\Gamma(V, {\mathfrak t}) \cup$ {0} is a 2-dimensional vector space over the prime field $\mathbb{F}_p$.

In some cases the 0 weight is actually missing in $V$. For instance, this happens when $L$ is the Hamiltonian algebra $H(2;\underline{1};\Phi(\tau))^{(1)}$ and $V$ is the adjoint module. On the other hand, in this case does there exist a nontrivial restricted irreducible module $V$ for $L_{[p]}$ such that the 0 weight is in $\Gamma(V, {\mathfrak t})$?