Geodesic
Action of biregular automorphisms of the affine plane on pairs of matrices
Izvestiya. Mathematics , Tome 33 (1989) no. 2, pp. 433-439

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A study is made of the natural action of the group of biregular automorphisms of the affine plane on pairs of square matrices and pairs of symmetric square matrices. It is proved that in the open set of pairs $(A,B)$ such that $A$ is semisimple and $A$ and $B$ have no common invariant subspaces, the commutator $[A,B]=AB-BA$ is the unique invariant in both cases. Bibliography: 3 titles. 
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     author = {I. V. Artamkin},
     title = {Action of biregular automorphisms of the affine plane on pairs of matrices},
     journal = {Izvestiya. Mathematics },
     pages = {433--439},
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     volume = {33},
     number = {2},
     year = {1989},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1989_33_2_a12/}
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I. V. Artamkin. Action of biregular automorphisms of the affine plane on pairs of matrices. Izvestiya. Mathematics , Tome 33 (1989) no. 2, pp. 433-439. http://geodesic.mathdoc.fr/item/IM2_1989_33_2_a12/