Jacobi field J along geodesic, satisfying additional condition $[J, \frac{\partial}{\partial r}] = 0.$ Following came up reading different articles and books:
Let $p$ be a point on a Riemannian manifold. For $x \in M \backslash Cut(p)$ let $\gamma$ be the minimal geodesic joining $p$ and $x$ parametrized by the distance, such that $\gamma(0) = p$, $\gamma(r) = x$.
Now let $X$ be a vector in $T_x M$ such that $\langle X, \frac{\partial}{\partial r} \rangle$ (x) = 0. 
Since $x$ is not a conjugate point of $p$, we can extend $X$ to a Jacobi field $\tilde{X}$ along $\gamma$ satisfying $\tilde{X}(\gamma(0)) = 0, \tilde{X}(\gamma(r)) = X$ and $[\tilde{X}, \frac{\partial}{\partial r}] = 0.$
My problem is the last part: Why does such a Jacobi field exist, that it commutes with $\frac{\partial}{\partial r}$?
More particular: Don't the first two conditions already determine $\tilde{X}$ uniquely? If yes, how to see that the commutator vanishes? If not, how can one defind such a Jacobi field? 
Any help is much appreciated. Thanks!
 A: OK, I looked at the reference you cited in your comment (Lectures on Differential Geometry by Schoen and Yau -- for anyone interested, this argument occurs on page 3). It turns out that they pulled a sneaky trick here, which they didn't explain very well.
Although they say "we can extend $X$ to a Jacobi field $\tilde X$ along $\sigma$," what they're actually doing is something more subtle. First they create a Jacobi field along $\sigma$ with the appropriate values at $\gamma(0)$ and $\gamma(r)$, which is possible because $\gamma(r)$ is inside the cut distance from $\gamma(0)$ and thus it's not a conjugate point to $\gamma(0)$.
But then, without saying so, they're also extending that Jacobi field to an actual vector field in a neighborhood of the image of $\sigma$, with the requirement that it commute with $\partial/\partial r$. That you can do so is not exactly a trivial observation, but it's well known to differential geometers. The simplest way to see this is to note that in geodesic polar coordinates $(r,\theta^1,\dots,\theta^{n-1})$ centered at $p$, the Jacobi fields along radial geodesics that vanish at $p$ and are orthogonal to $\partial/\partial r$ are exactly the vector fields of the form $J = a^1\partial/\partial \theta^1 + \dots + a^{n-1}\partial/\partial\theta^{n-1}$ where the coefficients $(a^1,\dots,a^{n-1})$ are constants. Verifying this boils down to a couple of basic observations about normal coordinates: (1) The Gauss lemma shows that each vector field $\partial/\partial \theta^i$ is orthogonal to $\partial/\partial r$; (2) Each such vector field $J$ is the variation field of a variation through radial geodesics of the form $\gamma_s(t) = (t,sa^1,\dots,sa^{n-1})$ in polar coordinates, and therefore is a Jacobi field.
And finally, the fact that the extended vector field commutes with $\partial/\partial r$ follows immediately from the fact that the $a^i$'s are constants.
