In thispaper we find many families in the product space ℍ2×ℝ of complete embedded, simply connected, minimal and surfaces with constant mean curvature H such that |H|≤1/2. We study complete surfaces invariant either by parabolic or by hyperbolic screw motions. We study the notion of isometric associate immersions. We exhibit an explicit formula for a Scherk-type minimal surface. We give a one-parameter family of entire vertical graphs of mean curvature 1/2. We prove a generalized Bour lemma that can be applied to ℍ2×ℝ,𝕊2×ℝ and to Heisenberg’s space to produce a family of screw motion surfaces isometric to a given one.

An example of folie à deux with an uncommon clinical picture is presented. It concerns two spinster sisters, with no past psychiatric history, who became simultaneously manic while on holiday. The “Principal” was clearly deluded; the “Associate” shared these abnormal beliefs. After recovery and eighteen months later, one became depressed and the other shared her suicidal ideas, but this time the roles of the “Principal” and “Associate” were reversed. The shortcomings of the ICD10 and DSM-IIIR in classifying this syndrome are highlighted.