Let $S$ satisfy $S^i_{jk} = S^{i}_{kj}$. What is the general solution of the differential equation
$$ \dot{w_i} = S^{i}_{jk} w_jw_k \quad (i= 1,2,3)? $$
(Summation notation is used.)
I'm fairly sure the general solution consists of sums of Jacobi elliptic functions, but I end up with too few degrees of freedom when I try a solution of the form $w_i(t) = a_i \text{ sn }v t + b_i \text{ cn } vt + c_i \text{ dn } vt$.