  Let $A$ be a transcendental entire function of finite order. We show that if
the differential equation $w''+Aw=0$ has two linearly independent solutions
with only real zeros, then the order of $A$ must be an odd integer or one half
of an odd integer. Moreover, $A$ has completely regular growth in the sense of
Levin and Pfluger. These results follow from a more general geometric theorem,
which classifies symmetric local homeomorphisms from the plane to the sphere
for which all zeros and poles lie on the real axis, and which have only
finitely many singularities over finite non-zero values.
