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In this paper we study properties of the functions which satisfy modular equations for infinitely many primes. The two main results are:
1) every such function is analytic in the upper half plane;
2) if such function takes the same value in two different points 
${{z}_{1}}$ and 
${{z}_{2}}$ then there exists an 
$f$-preserving analytic bijection between neighbourhoods of ${{z}_{1}}$ and ${{z}_{2}}$.
