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

Second order linear differential equations with a basis of solutions having only real zeros