Variation of algebraically integrable adjoint foliated structures

Given a canonical algebraically integrable foliation on a klt projective variety, we study the variation of the ample models of the associated adjoint foliated structures with respect to the parameter. When the foliation is of general type, we show the finiteness of ample models if the parameter is sufficiently close to $1$. When the ambient variety is of general type, we show the finiteness of ample models for all parameters. A key ingredient in our proof is the equivalence between the existence of minimal models and the termination of MMP with scaling for algebraically integrable adjoint foliated structures.