Let \((M,\nabla)\) be a compact affine manifold whose group of global affine transformations has non-solvable connected component of the identity \(\text{Aff }(M,\nabla)_0\). In this paper, the author associates the field \(K\) (real, complex, or quaternionic) and an affine \(K\)-transverse foliation \(\mathcal F_U\) to a finite cover \((\widetilde M,\widetilde\nabla)\) of \((M,\nabla)\). If \(p=\text{codim}_K\mathcal F>1\), then it is shown that \((\widetilde M,\widetilde\nabla)\) is a total space of a fibration over the projective space \(P^1_K(K^p)\). A partial classification of these manifolds is given. Also, the differentiable structures of 3- and 4-dimensional affine manifolds are determined (up to the finite cover) such that \(\text{Aff }(M,\nabla)_0\) is not solvable.