Schwarzschild geodesics I've found on Wikipedia that energy $E$ and angular momentum $L$ of a particle are conserved quantities in Schwarzschild metric. It's written:
$$L=mr^2 \frac {d\phi} {d\tau},$$
$$E=mc^2\left(1-\frac{r_s}{r}\right)\frac{dt}{d\tau}.$$
And, from the metric, it finds these results:
$$\left(\frac{d\phi}{d\tau}\right)^2=\frac{L^2}{m^2r^4}$$
$$\left(\frac{dr}{d\tau}\right)^2=\frac{E^2}{m^2c^2}-\left(1-\frac{r_s}{r}\right)\left(c^2+\frac{L^2}{m^2r^2}\right).$$
$$\left(\frac{dt}{d\tau}\right)^2=\frac{E}{\left(1-\frac{r_s}{r}\right)mc^2}$$
That I need is to get the results including also the $\theta$ coordinate. So I tried this:
$$p_\phi=mr^2 \frac {d\phi} {d\tau}\qquad\qquad p_\theta=mr^2 \frac {d\theta} {d\tau}$$
$$L^2=p_\theta^2 + \sin^2\theta\ p_\phi^2.$$
$E$ is the same?
$$\left(\frac{d\phi}{d\tau}\right)^2=\frac{p_\phi^2}{m^2r^4}\qquad\qquad \left(\frac{d\theta}{d\tau}\right)^2=\frac{p_\theta^2}{m^2r^4}.$$
$\frac{dr}{d\tau}$ is the same?
$\frac{dt}{d\tau}$ is the same?
But I'm not sure. However, is it possible to get a similar result adding the $\theta$ coordinate?
PS: I am a beginner about GR, so I don't know many things about it.
 A: Rotate the coordinates so that you reduce your problem to the 'standard' treatment with $\theta$ constant, solve it, and then rotate again the coordinates to get the solution in the original system.
The orbit is in the plane identified by the position ($\vec r$) and velocity ($\vec v$) vectors. This plane should be rotated so that it becomes the $xy$ plane. The vector normal to the orbit plane is $\hat n = (\vec r \times \vec v) / (rv)$, that should be rotated to become the $z$ axis - so the rotation axis is $\hat n \times \hat z$ and the rotation angle is $\arccos(\hat n \cdot \hat z)$
A: What you really want are two sections of wikipedia:
https://en.wikipedia.org/wiki/Schwarzschild_geodesics#Mathematical_derivations_of_the_orbital_equation
https://en.wikipedia.org/wiki/Schwarzschild_geodesics#Orbits_of_test_particles
It might be tough going for a while but understanding those will keep you from egregious errors.  Such that I (and others) have made.
I think some expansion of the question is appropriate.
A more correct question would be:
1)Given a mathematical description of a physical situation can we find invariants for certain types of orbits/paths/histories.
i.e. In the case: What are the invariants of geodesic paths/orbits in the Schwarzschild metric? BTW a fixed point with "time" (time is not a fixed coordinate for different coordinate systems) as the only variable is not a geodesic.
Then:
2)Can we identify these invariants with energy and momentum conservation?  
The first part is answered by a Theorem/construction/proof; inside of mathematical physics, it is not a given!  Although in our limited physics experience the conservation of energy and angular momentum have proven true and useful; in mathematical physics they must be reproved for separate situations.
As far as I know the questions are unresolved for general situations in GR.  
