Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following:

If $\gamma:[t_0,t_1]\rightarrow M$ is a smooth curve and is a geodesic, then in local coordinates we have $$ \ddot{\gamma}^\mu(t)+\Gamma^\mu_{\alpha\beta}(\gamma(t))\dot{\gamma}^\alpha(t)\dot{\gamma}^\beta(t)=0. $$ By defining $\dot{\gamma}(t)=v(t)$, this is a first-order system of ODEs: $$ \frac{dv^\mu}{dt}=-\Gamma^\mu_{\alpha\beta}v^\alpha v^\beta \\ \frac{d\gamma^\mu}{dt}=v^\mu. $$ Since this is an ODE for the variables $(x^1,...,x^n,v^1,...,v^n)$, this is a differential equation on the tangent bundle of $M$, $TM$, moreover, this is a first-order, homogenous differential equation on $TM$, so it is represented by a vector field, $G\in\Gamma(TTM)$, which I'll call the *geodesic flow*.

I also know that this vector field can be defined invariantly if we take $\gamma_{(p,v)}(t)=\exp_p(tv)$, eg. the maximal geodesic with initial point $\gamma_{(p,v)}(0)=p$ and $\dot{\gamma}_{(p,v)}(0)=v$ and we take $\bar{\gamma}_{p,v}(t)=(\gamma_{(p,v)}(t),\dot{\gamma}_{(p,v)}(t))$ the natural lift of the curve to $TM$, then $$G_{(p,v)}=\left.\frac{d}{dt}\bar{\gamma}_{(p,v)}\right|_{t=0}.$$ In local coordinates $(x,v)$ this obviously has the form $$ G_{(p,v)}=v^\mu\left.\frac{\partial}{\partial x^\mu}\right|_{p}-\Gamma^\mu_{\alpha\beta}(p)v^\alpha v^\beta\left.\frac{\partial}{\partial v^\mu}\right|_{v}. $$

I don't have any textbooks that deal with this any further however, so I have two questions.

**Question 1:** Given a smooth vector field $G\in\Gamma(TTM)$, what is the criteria for $G$ to be the geodesic flow of a linear connection? I can see from the local coordinate formula that the $x^\mu$ components contain only $v^\mu$ and the $v^\mu$ components are a quadratic form of the $v$ coordinates, but I am looking for invariant characterization. It is also clear to me that $G$ must not be vertical anywhere, but I don't think this is enough.

**Question 2:** If we are given $G$, how can I recover the *torsionless* connection? I mean, if I write up the local coordinate form, I can read off the connection coefficients and define the covariant derivative as $\partial_\mu V^\nu+\Gamma^\nu_{\mu\sigma}V^\sigma$ but I am once again looking for invariant characterization. A limit or $t$-derivative or something like that which reproduces the covariant derivative on the base if I know $G$.