Can an inviscid flow rotational? Potential Flow?

Hi all.
It seems to me that the origin of vorticity comes from viscosity, right?
So, if a flow is inviscid, can it be rotational?

For the example of a shear flow, U[y] = k*y, the vorticity a non-zero constant, k, everywhere, right? And we actually did not consider whether the flow is viscous or not when considering this shear flow. If the flow is inviscid, the vorticity is still constant and non-zero, right? So, it seems that an inviscid flow can still be rotational, right?

So, in the shear flow case, what is the difference between having the fluid being viscous and non-viscous? If the viscosity being zero but vorticity being non-zero, that means there is angular momentum (from non-zero vorticity) but no shear stress? Since the coefficient of viscosity is zero? It seems rather non-intuitive to me...We can have a zero shear stress but non-zero vorticity or angular momentum?

Also, why a potential flow must be inviscid? can it be just irrotational? Is that a viscous flow must be rotational?