A function \(X\in C^ 3(R)\), X’(t)\(\neq 0\) for \(t\in R\), is called the first kind dispersion of (q): \(y''=q(t)y\) \((q\in C^ 0(R))\) if it is a solution of the Kummer equation \(-X\prime''/2X'+3/4(X''/X')^ 2+X^{'2}q(X)=q(t)\) [see O. Borůvka, Linear differential transformations of the second order (1971; Zbl 0222.34002)]. The set of increasing dispersions of an oscillatory equation (q) constitutes a group \(L^+_ q\) relative to the composition of functions. Let (p), (q) be oscillatory equations, \(p\neq q\). Say that the group \(L^+_ p\cap L^+_ q\) is planar if there exists to every point \((t_ 0,x_ 0)\in R\times R\) a unique \(X\in L^+_ p\times L^+_ q\) such that \(X(t_ 0)=x_ 0\). It is proved that the group \(L^+_ p\cap L^+_ q\) is either trivial or infinite cyclic or planar.