Symmetric Division of Linear Ordinary Differential Operators

The symmetric product of two ordinary linear differential operators $L_1,L_2$ is an operator whose solution set contains the product $f_1{f}_2$ of any solution $f_1$ of $L_1$ and any solution $f_2$ of~$L_2$. It is well known how to compute the symmetric product of two given operators $L_1,L_2$. In this paper we consider the corresponding division problem: given a symmetric product $L$ and one of its factors, what can we say about the other factors?