Context-free languages are not closed under intersection.
Suppose $L_1, L_2 \in CF \setminus REG$ (i.e., $L_1,L_2$ are context-free but not regular).
Are there well-known theorems (and/or whole papers/research topics) that try to shape the sufficient/necessary conditions for $L_1,L_2$ to make $L_1 \cap L_2$ context free?