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Abstract

We describe QCH Kähler surfaces $(M,g,J)$ of generalized orthotoric type. We introduce a distinguished orthonormal frame on $(M,g)$ and give the structure equations for $(M,g,J)$. In the case when the opposite Hermitian structure $I$ is conformally Kähler and $(M,g,J)$ is not hyperkähler we integrate these structure equations and construct orthotoric Kähler surfaces in a new way. We also investigate the hyperkähler case. We prove in a simple way that if $(M,g,J)$ is a hyperkähler surface with a degenerate Weyl tensor $W^-$ (i.e. a QCH hyperkähler surface) then among all hyperkähler structures on $(M,g)$ there exists a Kähler structure $J_0$ such that $(M,g,J_0)$ is of Calabi type or of orthotoric type.