  For lc algebraically integrable foliations on klt varieties, we prove the
base-point-freeness theorem, the contraction theorem, and the existence of
flips. The first result resolves a conjecture of Cascini and Spicer, while the
latter two results strengthen a result of Cascini and Spicer by removing their
assumption on the termination of flips.
  Moreover, we prove the existence of the minimal model program for lc
algebraically integrable foliations on klt varieties and the existence of good
minimal models or Mori fiber spaces for lc algebraically integrable foliations
polarized with ample divisors on klt varieties. As a consequence, we show that
$\mathbb{Q}$-factorial klt varieties with lc algebraically integrable Fano
foliation structures are Mori dream spaces. We also show the existence of a
Shokurov-type polytope for lc algebraically integrable foliations.
