We consider germs of holomorphic vector fields at the origin in dimension 3 with non-isolated singularities that are tangent to a holomorphic foliation of codimension one, referred to as a 2-flag of foliations. Our focus lies on cases where this geometric structure originates from second-order ordinary differential equations. We completely comprehend the behavior of the singular sets associated with the foliations under consideration. Furthermore, we establish rigidity results that arise from the presence of a 2-flag associated with the second-order ordinary differential equation, and we present a classification result for those equations. Finally, we provide conditions for any n-th-order ordinary differential equation to admit a 2-flag of holomorphic foliations. Moreover, in such a case, we conclude that the equation will admit two functionally independent holomorphic first integrals.