  Horizonless spacetimes describing highly compact exotic objects with
reflecting (instead of absorbing) surfaces have recently attracted much
attention from physicists and mathematicians as possible quantum-gravity
alternatives to canonical classical black-hole spacetimes. Interestingly, it
has recently been proved that spinning compact objects with angular momenta in
the sub-critical regime ${\bar a}\equiv J/M^2\leq1$ are characterized by an
infinite countable set of surface radii, $\{r_{\text{c}}({\bar
a};n)\}^{n=\infty}_{n=1}$, that can support asymptotically flat static
configurations made of massless scalar fields. In the present paper we study
analytically the physical properties of ultra-spinning exotic compact objects
with dimensionless angular momenta in the complementary regime ${\bar a}>1$. It
is proved that ultra-spinning reflecting compact objects with dimensionless
angular momenta in the super-critical regime
$\sqrt{1-[{{m}/{(l+2)}}]^2}\leq|{\bar a}|^{-1}<1$ are characterized by a finite
discrete family of surface radii, $\{r_{\text{c}}({\bar
a};n)\}^{n=N_{\text{r}}}_{n=1}$, distributed symmetrically around $r=M$, that
can support spatially regular static configurations of massless scalar fields
(here the integers $\{l,m\}$ are the harmonic indices of the supported static
scalar field modes). Interestingly, the largest supporting surface radius
$r^{\text{max}}_{\text{c}}({\bar a})\equiv \text{max}_n\{r_{\text{c}}({\bar
a};n)\}$ marks the onset of superradiant instabilities in the composed
ultra-spinning-exotic-compact-object-massless-scalar-field system.
