  In arXiv:2401.00636 we introduced the notion of a periodic pencil of flat
connections on a smooth algebraic variety $X$. Namely, a pencil is a linear
family of flat connections
$\nabla(s_1,...,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i$, where $\lbrace
x_i\rbrace$ are local coordinates on $X$ and $B_{ij}: X\to {\rm Mat}_N$ are
matrix-valued regular functions. A pencil is periodic if it is generically
invariant under the shifts $s_j\mapsto s_j+1$ up to isomorphism. In this paper
we show that in characteristic $p$, the $p$-curvature operators $\lbrace
C_i,1\le i\le r\rbrace$ of a periodic pencil $\nabla$ are isospectral to the
commuting 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 of
arXiv: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, their
confluent limits, and equivariant quantum connections for conical symplectic
resolutions with finitely many torus fixed points. We also show that a periodic
pencil of connections always has regular singularites.
