Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures

Assuming the abundance conjecture in dimension d, we establish a non-algebraicity criterion of foliations: any log canonical foliation of rank ≤ d with ν≠κ is not algebraically integrable, answering question of Ambro--Cascini--Shokurov--Spicer. Under the same hypothesis, we prove abundance for klt algebraically integrable adjoint foliated structures of dimension ≤ d and show the existence of good minimal models or Mori fiber spaces. In particular, when $d=3$, all these results hold unconditionally. Using similar arguments, we solve a problem proposed by Lu and Wang on abundance of surface adjoint foliated structures that are not necessarily algebraically integrable.