A set of nonlinear coupled ODE

Question

I have reached a set of ODE as \begin{align} &\ddot{\vec{a}}(t)+\omega_0^2\frac{\cos{(b(t))}\sin{(a(t))}}{a(t)}\vec{a}(t)=0\\ &\ddot{b}(t)+\omega_0^2\cos{(a(t))}\sin{(b(t))}=0 \end{align} that describes the dynamics of a quantum operator $\hat{\Phi}(t)$ in the Heisenberg picture such that \begin{align} \hat{\Phi}(t)=\vec{a}(t).\vec{\sigma}+b(t)\mathbb{1} \end{align} In the equations above $a(t)$ represents the norm of $\vec{a}(t)=(a_x(t),a_y(t),a_z(t))$ and $\omega_0$ is the oscillation frequency. I need to solve this with the initial condition given as \begin{align} &a_x(0)=A \quad a_y(0)=0 \quad a_z(0)=0 \quad b(0)=0 \\ &\dot{a}_x(0)=0 \quad \dot{a}_y(0)=B \quad \dot{a}_z(0)=0 \quad \dot{b}(0)=0 \end{align} for nonzer $A$ and $B$. As far as I know there is no analytical solution for such an ODE recalling that even a 1D version of this $\ddot{a}(t)+\omega_0^2 \sin{(a(t))}=0$ can not be solved exactly. I wonder if this set of coupled nonlinear ODE has been studied in any context and if anyone has any intuition about its behavior.