  The KZ equations are differential equations satisfied by the correlation
functions (on the Riemann sphere) of two-dimensional conformal field theories
associated with an affine Lie algebra at a fixed level. They form a system of
complex partial differential equations with regular singular points satisfied
by the $n$-point functions of affine primary fields. In [SV1] the KZ equations
were identified with equations for flat sections of suitable Gauss-Manin
connections, and solutions of the KZ equations were constructed in the form of
multidimensional hypergeometric integrals. In [SV2] the KZ equations were
considered modulo a prime number $p$, and polynomial solutions of the KZ
equations modulo $p$ were constructed by an elementary procedure as suitable
$p$-approximations of the hypergeometric integrals. In this paper we address
the problem of whether all solutions of the KZ equations modulo $p$ are
generated by the $p$-hypergeometric solutions. We consider the first nontrivial
example of the KZ equations and demonstrate that, indeed, in this case, all
solutions of the KZ equations modulo $p$ stem from the $p$-hypergeometric
solutions.
