  Let $X$ be a toric Fano manifold and denote by $Crit(f_X) \subset
(\mathbb{C}^{\ast})^n$ the solution scheme of the corresponding Landau-Ginzburg
system of equations. For toric Del-Pezzo surfaces and various toric Fano
threefolds we define a map $L : Crit(f_X) \rightarrow Pic(X)$ such that
$\mathcal{E}_L(X) : = L(Crit(f_X)) \subset Pic(X)$ is a full strongly
exceptional collection of line bundles. We observe the existence of a natural
monodromy map $$ M : \pi_1(L(X) \setminus R_X,f_X) \rightarrow Aut(Crit(f_X))$$
where $L(X)$ is the space of all Laurent polynomials whose Newton polytope is
equal to the Newton polytope of $f_X$, the Landau-Ginzburg potential of $X$,
and $R_X \subset L(X)$ is the space of all elements whose corresponding
solution scheme is reduced. We show that monodromies of $Crit(f_X)$ admit
non-trivial relations to quiver representations of the exceptional collection
$\mathcal{E}_L(X)$. We refer to this property as the $M$-aligned property of
the maps $L: Crit(f_X) \rightarrow Pic(X)$. We discuss possible applications of
the existence of such $M$-aligned exceptional maps to various aspects of mirror
symmetry of toric Fano manifolds.
