Graphic around a surface

Dear McNeel,

how can I apply any kind of graphic around a surface?
In my file I have modeled a solid around which I would like to create continuous extruded designs, which I will then subtract using a Boolean command.

If I use the UnrollSrf command, unfortunately it generates a surface made of multiple parts, so the FlowAlongSrf command does not work correctly either.

I should mention that I am not interested in the top and bottom surfaces: the extruded graphic only needs to wrap around the solid.

How can I solve this problem?

Thank you.

Without an example is hard to understand the situation…
Maybe CreateUVCrv and then ApplyCrv?