By applying computer algebra tools mainly, Maple and C++, given the Dynkin diagram $\Delta = \mathbb{A}_n$, with n ≥ 2 vertices and the Euler quadratic form $q_\Delta : \mathbb{Z}^n \rightarrow \mathbb{Z}$, we study the problem of classifying mesh root systems and the mesh geometries of roots of Δ see Section 1 for details. The problem reduces to the computation of the Weyl orbits in the set $Mor_\Delta \subseteq \mathbb{M}_n\mathbb{Z}$ of all matrix morsifications A of qΔ, i.e., the non-singular matrices $A \in \mathbb{M}_n\mathbb{Z}$ such that i qΔv = v · A · vtr, for all $v \in \mathbb{Z}^n$, and ii the Coxeter matrix CoxA := -A · A-tr lies in $Gln,\mathbb{Z}$. The Weyl group $\mathbb{W}_\Delta \subseteq Gln, \mathbb{Z}$ acts on MorΔ and the determinant det $A \in \mathbb{Z}$, the order cA ≥ 2 of CoxA i.e. the Coxeter number, and the Coxeter polynomial $cox_At := dett \centerdot E \minus Cox_A \in \mathbb{Z}[t]$ are $\mathbb{W}_\Delta$-invariant. The problem of determining the $\mathbb{W}_\Delta$-orbits $\cal{O}rbA$ of MorΔ and the Coxeter polynomials coxAt, with $A \in Mor_\Delta$, is studied in the paper and we get its solution for n ≤ 8, and $A = [a_{ij}] \in Mor_{\mathbb{A}}_n$, with $\vert a_{ij} \vert \le 1$. In this case, we prove that the number of the $\mathbb{W}_\Delta$-orbits $\cal{O}rbA$ and the number of the Coxeter polynomials coxAt equals two or three, and the following three conditions are equivalent: i $\cal{O}rbA = \mathbb{O}rbA\prime$, ii coxAt = coxA't, iii cA · det A = cA' · det A'. We also construct: a three pairwise different $\mathbb{W}_\Delta$-orbits in MorΔ, with pairwise different Coxeter polynomials, if $\Delta = \mathbb{A}_{2m \minus 1}$ and m ≥ 3; and b two pairwise different $\mathbb{W}_\Delta$-orbits in MorΔ, with pairwise different Coxeter polynomials, if $\Delta = \mathbb{A}_{2m}$ and m ≥ 1.