  We present a method for constructing covariantly constant endomorphisms for
the mod $p$ equivariant quantum connection, using the quantum Steenrod power
operations of Fukaya and Wilkins. The example of the cotangent bundle of the
projective line is fully computed, and we discuss the relationship with the mod
$p$ solutions of trigonometric KZ equation recently constructed by Varchenko.
As a byproduct, we compute the first examples of quantum Steenrod operations
that are not a priori determined by ordinary Gromov--Witten theory and
classical Steenrod operations, which may be of independent interest.

Symplectic Geometry;Algebraic Geometry;Representation Theory