Consider the Lagrangian

$$L=A\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+B(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-C(\cos \theta)^{k}$$

where $A, B, C$ are positive constants and $k$ is a positive integer. Find three conserved quantities and show that $u=\cos \theta$ satisfies

$$\dot{u}^{2}=f(u)$$

where $f(u)$ is a polynomial of degree $k+2$ which should be determined.