$A$-analyticity of separatricies of foliations

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let $L$ be a foliation on $X$. We prove that if $L$ is closed under $p$-th powers for almost all primes $p$, then any $L$-invariant smooth formal curve is $A$-analytic. Building on prior work of Bost we obtain an algebraicity criterion for those curves.