Let S be a minimal compact complex surface with Betti numbers \(b_1(S)=1\) and \(b_2(S)\ge 1\) i.e. a compact surface in class VII \(_0^+\) . We show that if there exists a twisted logarithmic 1-form \(\theta \in H^0(S,\Omega ^1(\text{log} D)\otimes \mathcal L_\lambda )\) , where D is a non zero divisor and \(\mathcal L\in H^1(S,\mathbb C^\star )\) , then S is a Kato surface. It is known that \(\lambda \) is in fact real and we show that \(\lambda \ge 1\) and unique if S is not a Inoue-Hirzebruch surface. Moreover \(\lambda =1\) if and only if S is a Enoki surface. When \(\lambda >1\) these conditions are equivalent to the existence of a negative PSH function \(\hat \tau \) on the cyclic covering \(p:\hat S\to S\) of S which is PH outside \(\hat D:=p^{-1}(D)\) with automorphic constant being the same automorphy constant \(\lambda \) for a suitable automorphism of \(\hat S\) . With previous results obtained with V.Apostolov it suggests a strategy to prove the GSS conjecture.