On the Classification of Non-Homogeneous Solvable Lie Foliations

We study Lie foliations on compact manifolds whose transverse group is \emph{metabelian} (a natural generalization of the affine group $\GA$ considered in earlier work). We establish a complete classification of $\GA$-Lie foliations in dimension $5$, completing the work initiated in Dathe--Ndiaye. We then extend this analysis to foliations whose transverse group is a non-split metabelian Lie group, proving the existence of non-homogeneous Lie foliations with such groups in the smallest possible dimension. We introduce a new obstruction to homogeneity via the group cohomology $H^{2}(\Gamma,\mathbb{Z})$ of the holonomy group, and give exotic examples showing that non-polycyclicity of holonomy is not the only obstruction to homogeneity.