  We consider the following 2-D Schr\"{o}dinger-Newton equation
\begin{eqnarray*} \begin{cases} -\Delta u+u=w|u|^{p-1}u \\ -\Delta w=2 \pi
|u|^p \end{cases}\text{in} \; \mathbb{R}^2 \end{eqnarray*} for $ p \geq 2 $.
Using variational method with the Cerami compactness property, we prove the
existence of minimal action odd solutions. Also by carefully applying the
method of moving plane to a similar but more complex equation on the upper half
space, we prove these solutions are in fact axially symmetric. Our results
partially can be seen as the counterpart of results in paper \cite{GS} for the
2-D case, or the extension of the results \cite{CW} to the odd solution case.
