> Here are lots of difference surfaces found in the url > you given. Little bit confused with them.
There is one pseudospherical surface of revolution only and on them are super-imposed thickened tubular lines as geodesics and closely packed orthogonal trajectories that I call involutes. They are rotational copies, even if they appear to be one too many.
In literature heard that Hadamard investigated them a century ago, however seem not too complicated.