Deformations of Diophantine systems for quadratic forms of the root lattices $A_n$

The paper considers a method of deformation of Diophantine quadratic systems in the $n$-dimensional root lattices $A_n$, which allows one to obtain sections of matrix quadratic equations $Q[X]=A$ for quadratic forms $Q$ of the lattices $A_n$ and formulas for the number of form representations by the corresponding sections.