I am looking for examples of 2 and 3dimensional flat spin manifolds with Euclidean and Lorentzian signatures, which admit parallel spinors and the dimension of the space of the parallel spinors is equal to 1. At the moment I have examples only for the Lorentzian case:$\mathbb{R}^{1,1}_{*}/(Z_{2})$ (for dimension 2) and $\mathbb{R}^{1,1}_{*}/(Z_{2}) \times \mathbb{R}$ (for dimension 3). Here $\mathbb{R}^{1,1}_{*}$ means $\mathbb{R}^{1,1}\setminus \{(0,0)\}$ and the generator of $Z_2$ acts as $x \mapsto (x)$. However, I am not sure if it is correct.
