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Urata showed that a pointed compact hyperbolic variety admits only finitelymany maps from a pointed curve. We extend Urata's theorem to the setting of(not necessarily compact) hyperbolically embeddable varieties. As anapplication, we show that a hyperbolically embeddable variety over a numberfield $K$ with only finitely many $\mathcal{O}_{L,T}$-points for any numberfield $L/K$ and any finite set of finite places $T$ of $L$ has, in fact, onlyfinitely many points in any given $\mathbb{Z}$-finitely generated integraldomain of characteristic zero. We use this latter result in combination withGreen's criterion for hyperbolic embeddability to obtain novel finitenessresults for integral points on symmetric self-products of smooth affine curvesand on complements of large divisors in projective varieties. Finally, we use apartial converse to Green's criterion to further study hyperbolic embeddability(or its failure) in the case of symmetric self-products of curves. As aby-product of our results, we obtain the first example of a smooth affineBrody-hyperbolic threefold over $\mathbb{C}$ which is not hyperbolicallyembeddable.

Urata's theorem in the logarithmic case and applications to integral points