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We construct a connected, irreducible component of the moduli space ofminimal surfaces of general type with $p_g=q=2$ and $K^2=5$, which containsboth examples given by Chen-Hacon and the first author. This component isgenerically smooth of dimension 4, and all its points parametrize surfaceswhose Albanese map is a generically finite triple cover.

On surfaces with p_g=q=2, K^2=5 and Albanese map of degree 3