  In this paper, we study the $\mu$-variant of the periodic $b$-equation and
show that this equation can be realized as a metric Euler equation on the Lie
group $\Diff^{\infty}(\S)$ if and only if $b=2$ (for which it becomes the
$\mu$-Camassa-Holm equation). In this case, the inertia operator generating the
metric on $\Diff^{\infty}(\S)$ is given by $L=\mu-\partial_x^2$. In contrast,
the $\mu$-Degasperis-Procesi equation (obtained for $b=3$) is not a metric
Euler equation on $\Diff^{\infty}(\S)$ for any regular inertia operator
$A\in\mathcal L_{\text{is}}^{\text{sym}}(C^{\infty}(\S))$. The paper
generalizes some recent results of [J. Escher and B. Kolev, DOI
10.1007/s00209-010-0778-2], [J. Escher and J. Seiler, J. Math. Phys. 51 (2010),
053101.1-053101.6] and [B. Kolev, Wave Motion 46 (2009), 412-419].
