SU(1) is the trivial group. SU(n) is the group of unitary nxn matrices with determinant 1. There is only one 1x1 unitary matrix with determinant 1, and in fact there is only one 1x1 matrix with determinant 1 period, namely, the 1x1 matrix whose only entry is the number 1.

So, to associated something with a symmetry group of SU(1) would be effectively the same as not associating it with a symmetry group.