Surfaces of fundamental type with geometric genus 2 and $c_1^2|X|=1$

In [1] E. Bombieri showed that $|4K|$ always yields a holomorphic map for surfaces of fundamental type and that $|3K|$ does not yield a holomorphic map for such surfaces with $p_g=2$ and $c_1^2|X|=1$. In this note we prove the existence of such surfaces and give a complete description of them. We prove that Torelli's local theorem is true, i.e., that the mapping of periods from the space of moduli into the space of periods is étale; we calculate the number of moduli and we show that the space of moduli is nonsingular.