A surface is bounded by four lines parametrised as $(x,y,z)=$

$$ (0,u,- 1), (-1<u<1); \, (0,u,1), (-1<u<1); $$

$$(\cos v, \sin v, 2 v/ \pi), (- \pi/2 < v< \pi/2); \, (-\cos v, -\sin v, 2v /\pi), (-\pi/2,< v < \pi/2); \,$$ It is required to find parametrization for constant $K$ surfaces whose

1. K= -1
2. K= 0
3. K= +1

The Dini surface does not meet requirement of a helix border for case 1. Neither Mean curvature H =0 helicoid of varying K satisfies this case 1.

Untwisted constant $ H$ CMC surfaces catenoid, cylinder, sphere of soap films form across two concentric circular tube edges of radius 1 rotated on x-axis.They have respectively their ODE connecting principal curvatures as:

$$ \kappa_1 + \kappa_ 2 = T,$$

where constant surface tension T can take $ -1, 0, +1 $ values.Their twisted surface parametrization is now sought, thanks for your help here.