Gorenstein stable surfaces with Kₓ² = 1 and pg>0

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In this paper we consider Gorenstein stable surfaces with K²ₓ=1 and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with Kₓ²=1; for pg=2 this leads to a rough stratification of the moduli space. Explicit non-Gorenstein examples show that we need further techniques to understand all possible degenerations.