Special birational structures on non-Kähler complex surfaces

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We investigate the following conjecture: all compact non-Kähler complex surfaces admit birational structures. After Inoue-Kobayashi-Ochiai, the remaining cases to study are essentially surfaces in class VII₀+. In case of Kato surfaces with a cycle and one branch of rational curves we show that they have a special birational structure given by new normal forms of contracting germs in Cremona group Bir(P²(C)). In particular all surfaces S with GSS and 0<b₂(S)<4 admit a birational structure. From the existence of a special birational structure we deduce meromorphic mappings from the universal cover of S to the projective plane which blow down an infinite number of rational curves.