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We consider the KZ differential equations over $\mathbb C$ in the case, whenits multidimensional hypergeometric solutions are one-dimensional integrals. Wealso consider the same differential equations over a finite field $\mathbbF_p$. We study the space of polynomial solutions of these differentialequations over $\mathbb F_p$, constructed in a previous work by V. Schechtmanand the author. The module of these polynomial solutions defines an invariantsubbundle of the associated KZ connection modulo $p$. We describe the algebraicequations for that subbundle and argue that the equations correspond to highestweight vectors of the associated $\hat{sl}_2$ Verma modules over the field$\mathbb F_p$.

An invariant subbundle of the KZ connection mod $p$ and reducibility of $\hat{sl}_2$ Verma modules mod $p$