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For the geometry of oriented $(2, 3, 5)$ distributions $(M, {\mathbf D})$,which correspond to regular, normal parabolic geometries of type$(\mathrm{G}_2, P)$ for a particular parabolic subgroup $P < \mathrm{G}_2$, wedevelop the corresponding tractor calculus and use it to analyze the first BGGoperator $\Theta_0$ associated to the $7$-dimensional irreduciblerepresentation of $\mathrm{G}_2$. We give an explicit formula for the normalconnection on the corresponding tractor bundle and use it to derive explicitexpressions for this operator. We also show that solutions of this operator areautomatically normal, yielding a geometric interpretation of $\ker \Theta_0$:For any $(M, {\mathbf D})$, this kernel consists precisely of the almostEinstein scales of the Nurowski conformal structure on $M$ that ${\mathbf D}$determines.  We apply our formula for $\Theta_0$ (1) to recover efficiently some knownsolutions, (2) to construct a distribution with root type $[3, 1]$ with anonzero solution, and (3) to show efficiently that the conformal holonomy of aparticular $(2, 3, 5)$ conformal structure is equal to $\mathrm{G}_2$.

The almost Einstein operator for $(2, 3, 5)$ distributions