This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.
There were 368 solutions found in 82.589 seconds (with a Bezout bound of 4608).
References:
W. Boege, R. Gebauer, and H. Kredel: "Some examples for solving systems of algebraic equations by calculating Groebner bases", J. Symbolic Computation, 2:83-98, 1986.
C. Butcher: "An application of the Runge-Kutta space". BIT, 24, pages 425–440, 1984.
This example is from the POSSO test suite available by anonymous ftp from the site gauss.dm.unipi.it, from the directory pub/posso.
See also: http://homepages.math.uic.edu/~jan/Demo/butcher.html
i1 : butcher(QQ)
2 1 1 2 2 2 3 2 1 4
o1 = {z*u + y*v + t*w - w - -w - -, z*u + y*v - t*w + w + w - -t + -w,
2 2 3 3
------------------------------------------------------------------------
2 3 1 2 1 2 3 3 3 4 3 3
- t*w + w + z*v*x - -t*w + w - -t + -w, z*u + y*v + t*w - w - -w
2 6 3 2
------------------------------------------------------------------------
5 2 1 1 3 4 1 2 3 3 1 7 2
+ t*w - -w - -w - -, t*w - w + z*u*v*x + -t*w - -w + -t*w - -w -
2 4 4 2 2 2 4
------------------------------------------------------------------------
3 1 3 4 2 2 3 3 2 7 2 1 1 3
-w - -, t*w - w + z*v x + t*w - -w + -t*w - -w - --w - --, - t*w +
8 8 2 3 6 12 12
------------------------------------------------------------------------
4 2 3 3 1 13 2 7 1
w - t*w + -w - -t*w + --w + --w + --}
2 3 12 24 24
o1 : List
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The object butcher is a method function.