Geodesic
Deformations of Diophantine systems for quadratic forms of the root lattices~$A_n$
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 5-14

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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. 
@article{ZNSL_2004_314_a0,
     author = {N. V. Budarina},
     title = {Deformations of {Diophantine} systems for quadratic forms of the root lattices~$A_n$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--14},
     publisher = {mathdoc},
     volume = {314},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_314_a0/}
}
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N. V. Budarina. Deformations of Diophantine systems for quadratic forms of the root lattices~$A_n$. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 5-14. http://geodesic.mathdoc.fr/item/ZNSL_2004_314_a0/