This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 10 solutions found in 1.447 seconds (with a Bezout bound of 243).
Reference: "Homotopy method for general lambda-matrix problems" by M. Chu, T.Y. Li and T. Sauer (pages 528-536).
See also: http://homepages.math.uic.edu/~jan/Demo/geneig.html.
i1 : geneig(QQ)
2 2 2 2 2
o1 = {- 10x x + 2x x - x x + x x + 3x x + x x + 2x x + x x + 2x x +
1 6 2 6 3 6 4 6 5 6 1 6 2 6 3 6 4 6
------------------------------------------------------------------------
2 2 2 2
x x + 10x + 2x - x + 2x - 2x , 2x x - 11x x + 2x x - 2x x +
5 6 1 2 3 4 5 1 6 2 6 3 6 4 6
------------------------------------------------------------------------
2
x x + 2x x + x x + 2x x + x x + 3x x + 2x + 9x + 3x - x - 2x ,
5 6 1 6 2 6 3 6 4 6 5 6 1 2 3 4 5
------------------------------------------------------------------------
2 2 2 2 2
- x x + 2x x - 12x x - x x + x x + x x + 2x x - 2x x - 2x x -
1 6 2 6 3 6 4 6 5 6 1 6 2 6 4 6 5 6
------------------------------------------------------------------------
2 2 2 2 2
x + 3x + 10x + 2x - x , x x - 2x x - x x - 10x x + 2x x + 2x x
1 2 3 4 5 1 6 2 6 3 6 4 6 5 6 1 6
------------------------------------------------------------------------
2
+ x x - 2x x + 2x x + 3x x + 2x - x + 2x + 12x + x , 3x x +
2 6 3 6 4 6 5 6 1 2 3 4 5 1 6
------------------------------------------------------------------------
2 2 2 2
x x + x x + 2x x - 11x x + x x + 3x x - 2x x + 3x x + 3x x -
2 6 3 6 4 6 5 6 1 6 2 6 3 6 4 6 5 6
------------------------------------------------------------------------
2x - 2x - x + x + 10x , x + x + x + x + x - 1}
1 2 3 4 5 1 2 3 4 5
o1 : List
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The object geneig is a method function.