Codimension one foliations on adjoint varieties

In this paper, we classify codimension one foliations on adjoint varieties with most positive anti-canonical class. We show that on adjoint varieties with Picard number one, these foliations are always induced by a pencil of hyperplane sections with respect to their minimal embedding. For adjoint varieties of Picard number two, there is more than one component of such foliations, and we describe each of them. As a tool for understanding these foliations, we introduce the concept of the degree of a foliation with respect to a family of rational curves, which may be of independent interest.