Log Canonical Minimal Model Program for corank one foliations on Threefolds

If $(X, \mcF, \D)$ is a projective rank two foliated log canonical triple such that $(X,B)$ is klt for some $0 \leq B \leq \D$, we show that we can run a $(K_\mcF +\Delta)$-MMP and any such MMP terminates with either a minimal model or Mori fiber space. Next, we establish a Bertini type lemma and adjunction for generalized foliated quadruples. Using these, we extend the full log canonical MMP to the setting of rank two NQC generalized foliated quadruples. Finally, we apply the generalized MMP to study the relation between different minimal models, namely, any two minimal models of a given foliated log canonical triple can be connected by a sequence of flops and in the boundary polarized case, the minimal models are good and only finitely many in number.