  In the doped topological insulator Sr$_x$Bi$_2$Se$_3$, a pronounced in-plane
two-fold symmetry is observed in electronic properties below the
superconducting transition temperature $T_c \sim$ 3 K, despite the three-fold
symmetry of the observed $R\bar{3}m$ space group. The axis of two-fold symmetry
is nominally pinned to one of three rotational equivalent directions and
crystallographic strain has been proposed to be the origin of this pinning. We
carried out multimodal synchrotron diffraction and resistivity measurements
down to $\sim$0.68 K and in magnetic fields up to 45 kG on a single crystal of
Sr$_{0.1}$Bi$_2$Se$_3$ to probe the effect of superconductivity on the
crystallographic distortion. Our results indicate that there is no in-plane
crystallographic distortion at the level of $1x10^{-5}$ associated with the
superconducting transition. These results further support the model that the
large two-fold in-plane anisotropy of superconducting properties of
Sr$_x$Bi$_2$Sr$_3$ is not structural in origin but electronic, namely it is
caused by a nematic superconducting order parameter of Eu symmetry.
