We consider the following 4 dimensional open manifold $$M=Gl(2,\mathbb{R})\setminus \{\lambda I_2 \mid \lambda \in \mathbb{R}\}$$ where $I_2$ is the identity matrix.

We consider the $2$ dimensional foliation $\mathcal{F}$ of $M$ tangent to the vector fields $X(A)=A, Y(A)=A^2$ for $A\in M$.

Is there a leaf of this foliation with nontrivial holonomy?

Is the leaf space of this foliation, a Hausdorff space?

Is there a Riemannian metric on $M$ such that the leaves of the above foliations are totally geodesic immersed submanifolds?