Surfaces with quadratic support function of harmonic type

In this paper, we study oriented surfaces S in ${\mathbb{R}}^3$, called Surfaces with quadratic support function of harmonic type (in short HQSF-surfaces), these surfaces generalize the QSF-surfaces. We obtain a Weierstrass type representation for the HQSF-surfaces which depends on three holomorphic functions. Moreover, we classify the HQSF-surfaces of rotation.