Unfortunately, it's look like the situation is still the same described by Kjell, I don't know software good enough for determining game solutions, inclusive Nash equilibrium solutions. I have tried to use some years ago GAMS, but it was good for antagonistic games, not for polymatrix ones.
The package written in 1991 by John Dickaut still works with some warnings. It doesn't specify all equilibra, but only the ones I named reference equilibria. In the example presented there, it gives all three solutions.
But, what must the user know in the following example, obtained from precedent one by modifying a single element?
The set of all equilibria consists of one distinct (isolated) point and one interval. Such type of situations appears in more complicated form when the the number of strategies is greater... For example, for three person 2x2x2 gamed the set of all equilibria may consists of surfaces defined by nonlinear forms.
To generate pictures above, I have used the demonstration "Nash Equilibrium Sets in Dyadic Bimatrix Mixed-Strategy Games".
The problem of computing a Nash equilibrium in two-player game is PPAD-complete
(PPAD is an abbreviation for Polynomial Parity Argument for Directed graphs, see:
Papadimitriou, C. H., On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence, Journal of Computer and System Sciences, No. 48(3), 1994, pp. 498-532.
Daskalakis, C., Goldberg, P.W., and Papadimitriou, C.H. The complexity of computing a Nash equilibrium, Communications of the ACM, 52(2), 2009, pp. 89-97.).
