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The main goal of this article is to prove a second main theorem for entire curves into Hilbert modular surfaces. As an application of our main theorem, we obtain a condition ensuring that entire curves in a Hilbert modular surface of general type are contained in the exceptional divisors. We also show a second main theorem for a simple normal crossing divisor which is tangent to a holomorphic distribution of codimension one on a smooth projective algebraic manifold.

The second main theorem for entire curves into Hilbert modular surfaces