I was trying to derive equation of motion for test particle around a Kerr black hole. My work is as follows:
The Kerr metric is as follows
$$ \mathrm ds^2 = -\left(1-\dfrac{2Mr}{\rho^2}\right)\mathrm dt^2-\dfrac{4Ma}{\rho^2}\sin^2( \theta)\,\mathrm dt\,\mathrm d\phi+\dfrac{\rho^2}{\Delta}\,\mathrm dr^2+\rho^2\mathrm d\theta^2+\left((r^2-a^2)\sin^2(\theta)+\dfrac{2Mra^2}{\rho^2}\sin^4(\theta)\right)\mathrm d\phi^2$$
where \begin{align} \Delta&\triangleq r^2-2Mr+a^2\\ \rho^2&\triangleq r^2+a^2\cos^2(\theta) \end{align}
4-velocity vector is given as
$$u=(u^t,u^r,u^\theta,u^\phi)$$
and the basis killing vectors for the Kerr metric are given as follows $$\xi=(1,0,0,0)$$ $$\eta=(0,0,0,1)$$
hence, the conserved quantities are (say) $$\xi\cdot u=e$$ $$\eta\cdot u=l$$
which after little simplification is $$g_{tt}u^t+g_{t\phi} u^\phi = e$$ $$g_{\phi t}u^t+g_{\phi \phi} u^\phi = l$$
which can be solved for $u^t$ and $u^\phi$, both of them will be functions of $\theta$ and $\phi$.
Hence we have $$\boxed{u^t=F(r,\theta;e,l) \equiv F }$$ $$\boxed{u^\phi=G(r,\theta;e,l) \equiv G }$$
Third invariant quantity that can be created is by using the fact that for particles with $m\neq 0$ $u^\alpha u_\alpha =-1$
$$\Rightarrow u^\alpha u_\alpha = g_{tt}(u^t)^2+g_{rr}(u^{r})^2+g_{\phi\phi}(u^\phi)^2+g_{\theta \theta}(u^{\theta})^2+g_{t\phi}u^tu^\phi=-1$$
$$\Rightarrow g_{rr}(u^{r})^2+g_{\theta\theta}(u^\theta)^2+[g_{tt}F^2+g_{\phi\phi}G^2+g_{\phi t}FG] = -1$$
Renaming the terms in bracket as, say, $H(r,\theta; e,l) \equiv H$, hence
$$\boxed{ g_{rr}(u^{r})^2+g_{\theta\theta}(u^\theta)^2 + H = -1}$$
Now, we have got $3$ equations of motion but I don't understand how to integrate it (numerically or analytically). If we would have had $1$ more constraint involving either $r$ or $\theta$ we could have easily separated the equations and then integrated it but that it not the case as no other constraints are available (?).
So, just having these 3 equations is it possible to find the trajectory or do we require something more? If yes, then please provide what I am missing here to continue the calculation.
