  We study bifurcations of symmetric elliptic fixed points in the case of
\emph{p}:\emph{q} resonances with odd $q\geq 3$. We consider the case where the
initial area-preserving map $\bar z =\lambda z + Q(z,z^*)$ possesses the
central symmetry, i.e. is invariant under the change $z\to -z$, $z^*\to -z^*$.
We construct normal forms for such maps in the case $\lambda = e^{i 2\pi
\frac{p}{q}}$, where $p$ and $q$ are mutually prime integer numbers, $p\leq q$
and $q$ is odd, and study local bifurcations of the fixed point $z=0$ in
various settings. We prove the appearance of garlands consisting of four
$q$-periodic orbits, two orbits are elliptic and two orbits are saddle, and
describe the corresponding bifurcation diagrams for one- and two-parameter
families. We also consider the case where the initial map is reversible and
find conditions when non-symmetric periodic orbits of the garlands are
non-conservative (compose symmetric pairs of stable and unstable orbits as well
as area-contracting and area-expanding saddles).
