I tried with giac, I get a correct answer for the linear system, but the jordan/eigenvalues/eigenvects command fails unless you run trig2exp before. This is a dilemna for any symbolic computation program: how should one consider trigonometric expressions? The most efficient is to consider them as algebraically independant, I guess this is what happens in Maxima in the solver (while the eigenvalues are computed using Cardan formulae without trigonometric simplifications). That means you can probably not expect a workaround, because this is not a bug, it is a limitation or a tradeoff. At some point the user must understand how things work and do some operations manually, like it was partially done by searching the eigenspace for eigenvalue 1.
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