Is  there a  connection on $\mathbb{R}^2 \setminus \{0\}$ for  which  all  operators of  parallel transports are in the  form $$\begin{pmatrix}a&-b\\b&a \end{pmatrix}$$

but the parallel transport along  circles with  center at origin depends  on the  radius of the  circle. That is  two  different  circle  have  different  parallel transports.

This  question is    motivated  by [this  paper](https://www.duo.uio.no/bitstream/handle/10852/43009/1995-47.pdf?sequence=1&isAllowed=y).