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One can find a solution of the form $x=y=z$, namely, $x=2$arccot$(\exp (-t-a))$ with the free parameter $a$. Of course, there should be more. Note also the symmetries of the problem: For any solution $(x,y,z)$, also $(-x,-y,-z)$ is a solution, as is $(x+2k\pi,y+2l\pi,z+2m\pi)$, with arbitrary integers $k,l,m$.