The Pick invariant \(J\) is one of the most important invariants in affine hypersurface theory. On locally strongly convex hypersurfaces \(J\) measures the deviation from quadrics. Relatively little is known about the local behaviour of \(J\). The paper contributes to this topic and continues the investigation of locally strongly convex surfaces with constant Pick invariant [see also \textit{F. Dillen}, \textit{A. Martinez}, \textit{F. Milán}, \textit{F. G. Santos} and \textit{L. Vrancken}, Result. Math. 20, 622-642 (1991)].