Let $v_1, v_2, v_3 \in \mathbb{R}^3$ and let $M$ be some symmetric matrix. I have the following system of ODEs:
$$ \frac{dv_i}{dt}=v_i\times b $$
for $i=1,2,3$ and:
$$ b_a = v_b^TMv_c$$
for $(a,b,c)=(1,2,3),(2,3,1),(3,1,2)$.
Suppose that $v_1,v_2,v_3$ are orthonormal at time $t=0$. As $M$ is symmetric, it has an orthonormal basis of eigenvectors. I know that a steady state is reached when $v_1,v_2,v_3$ are the eigenvectors of $M$. Is there a way of solving this system for all time $t>0$?