I've been working on a project about the dynamics of spins and I've encountered the following system of ODEs which I'm unable to solve: $$\frac{dS_1}{dt} = S_1 \times S_2 + \gamma S_1 \times (S_1 \times S_2)$$ $$\frac{dS_2}{dt} = S_2 \times S_1 + \gamma S_2 \times (S_2 \times S_1).$$
What I did was to set $u = S_1 \times S_2,$ then differentiate to get $\frac{du}{dt} = \frac{dS_1}{dt} \times S_2 + S_1 \times \frac{dS_2}{dt}.$ Making the substitutions and after some calculations I arrived at $$\frac{du}{dt} = u \times (S_1 + S_2),$$ and I'm stuck at this point. I forgot to mention that $S = (S_x, S_y, S_z)$ with $\|S\| = 1$ and $\gamma >0.$
Any help to find a solution would be much appreciated. Thank you!