  Let $X$ be a manifold with a bi-Poisson structure $\{\eta^t\}$ generated by a
pair of $G$-invariant symplectic structures $\omega_1$ and $\omega_2$, where
the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for
this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the
submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra
equal to the Lie algebra $\mathfrak{h}$ of $H$ and with the stabilizer group
conjugated to $H$ in $G$. We prove that the pair of symplectic structures
$\omega_1|_{X^r_\mathfrak{h}}$ and $\omega_2|_{X^r_\mathfrak{h}}$ generates an
$N(H^0)/H^0$-invariant bi-Poisson structure on $X^r_\mathfrak{h}$, where
$N(H^0)$ is the normalizer in $G$ of the identity component $H^0$ of $H$. The
action of $\widetilde G=N(H^0)/H^0$ on $X^r_\mathfrak{h}$ is locally free and
proper and, moreover, the spaces $A^G$ of $G$-invariant functions on $X$ and
$A^{\widetilde G}$ of $\widetilde G$-invariant functions on $X^r_\mathfrak{h}$
can be canonically identified and therefore the bi-Poisson structure
$\{(\eta^t)'\}$ induced on $A^G\simeq A^{\widetilde G}$ can be treated as the
reduction with respect to a {\em locally free} action of a Lie group which
essentially simplifies the study of $\{(\eta^t)'\}$.
