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In arXiv:2401.00636 we introduced the notion of a periodic pencil of flatconnections on a smooth algebraic variety $X$. Namely, a pencil is a linearfamily of flat connections$\nabla(s_1,...,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i$, where $\lbracex_i\rbrace$ are local coordinates on $X$ and $B_{ij}: X\to {\rm Mat}_N$ arematrix-valued regular functions. A pencil is periodic if it is genericallyinvariant under the shifts $s_j\mapsto s_j+1$ up to isomorphism. In this paperwe show that in characteristic $p$, the $p$-curvature operators $\lbraceC_i,1\le i\le r\rbrace$ of a periodic pencil $\nabla$ are isospectral to thecommuting endomorphisms $C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}$, where$B_{ij}^{(1)}$ is the Frobenius twist of $B_{ij}$. Using the results ofarXiv:2401.00636, this allows us to compute the eigenvalues of the$p$-curvature for many important examples of pencils of flat connections,including Knizhnik-Zamolodchikov, Dunkl, and Casimir connections, theirconfluent limits, and equivariant quantum connections for conical symplecticresolutions with finitely many torus fixed points. We also show that a periodicpencil of connections always has regular singularites.

$p$-curvature of periodic pencils of flat connections