Punctual structures and punctual categoricity

A punctual algebraic structure $A$ (i.e., primitive recursive structure on the universe $\omega$) is punctually categorical if for every its punctual copy $B$ there is an isomorphism from $A$ onto $B$ which is primitive recursive together with the inverse.
Unexpectedly, there is an dichotomy for this notion: every punctually categorical structure is either finitely generated, or locally finite. This dichotomy also holds for the structures which have a degree of punctual categoricity.
For the finitely generated structures we can describe the possible degrees of punctual categoricity. We also have some partial results relating degrees of punctual categoricity of locally finite structures.