Optimal Synthesis in a Radially Symmetric Grushin Space

We study the geometry of ${\mathbb{R}}^3$ equipped with a rotationally invariant Carnot-Carthéodory metric obtained by weighting motion in the $z$-direction by a function $f(r)$ of the cylindrical radius. When $f$ vanishes only at $r=0$, the space exhibits a Grushin–type singularity along the vertical axis. We provide sufficient conditions on $f$ ensuring a Grushin–like structure and describe the full optimal synthesis at singular points. For Riemannian points, we propose a candidate cut time determined by a discrete symmetry of the Hamiltonian flow. In the integrable case $f(r)=r$, we prove that this candidate coincides with the true cut time and give an explicit description of the cut locus.