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Let $M$ be a two-dimensional manifold and let $\mathcal C$ be a family of smooth paths on $M$. How to understand whether this family is actually a family of (possibly reparametrized) geodesics of some affine connection?

In this paper by V. Matveev it is written that this problem is classical and the answer was known even to Sophus Lee. Matveev writes that in order to answer to this question one must construct an appropriate ODE $y''(x) = f(x,y(x),y'(x))$ and check whether the right hand side is a third degree polynomial in $y'$ or not.

My question is whether there is some classical book where this problem is considered in detail and the mentioned results are presented?

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