Let $c$ be a $W^{1,2}$-curve into a (compact Riemannian) manifold $Q,$ defined on some open interval $I$. Let $t_0\in I$ and $\xi_0\in T_{c(t_0)}Q$ be arbitrary. I am looking for a citeable reference for the following statement: There is a unique $W^{1,2}$-vector field $\xi$ along $c$ satisfying

(i) $\nabla_{\dot{c}}\xi=0$ a.e. on $I$

(ii) $\xi(t_0)=\xi_0.$

Notice that the statement above could be deduced from the standard Picard-Lindelöf theorem if $c$ was sufficiently regular (i.e. e.g. $c$ was $C^1$).