A circle quotient of a $G_2$ cone

A study is made of $R^6$ as a singular quotient of the conical space $R^{+}\times C{P}^3$ with holonomy $G_2$ with respect to an obvious action by $U(1)$ on $C{P}^3$ with fixed points. Closed expressions are found for the induced metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of $SO(3)$ on $R^6$, which can be used effectively to describe the resulting geometrical features.