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

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

$$(\cos v, \sin v, 2 v/ \pi), (- \pi/2 < v< \pi/2);  \, (-\cos v, -\sin v, 2v /\pi), (-\pi/2,< v < \pi/2);  \,$$
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 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.