Is this system of ODE's exactly solvable?

$\displaystyle {\sin^2(\theta(t))\dot{\phi}(t)=C}\brace{\sin(\the ta(t)\cos(\theta(t))\dot{\phi}^2(t)=\ddot{\theta}( t)}$

Notice that above the $\displaystyle \phi$ is a dot which of course is meant to represent $\displaystyle \frac{\partial}{\partial t}$

If anyone's curious it arose from trying to find the extremals of the functional $\displaystyle J\left(\theta,\phi\right)=\int\left(\left\|\frac{\ partial \vec{x}}{\partial t}\right\|^2-U(t)\right)dt$ where in this case $\displaystyle U(t)=0$ and we're assuming that $\displaystyle \vec{x}\in\mathbb{S}^2$