  By considering spectral triples on $S^{2}_{\mu, c}$ ($c>0$) constructed by
Chakraborty and Pal (\cite{chak_pal}), we show that in general the quantum
group of volume and orientation preserving isometries (in the sense of
\cite{goswami2}) for a spectral triple of compact type may not have a
$C^*$-action, and moreover, it can fail to be a matrix quantum group. It is
also proved that the category with objects consisting of those volume and
orientation preserving quantum isometries which induce $C^*$-action on the
$C^*$ algebra underlying the given spectral triple, may not have a universal
object.

Operator Algebras;Quantum Algebra