Integral Curve of the vector field

Question

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos \theta) $. How to compute the integral curve of $X$ that passes through the point $(\theta, \phi) = (0,0)$ at $t=0$? How the definition fits to compute the integral curve? Alongside what is the parallel transport of the vector $Y= (-r\cos \theta \sin \phi, r\sin \theta \cos \phi, 0)$ along the integral curve of $X$? Thanks.