Symplectic rational $G$-surfaces and equivariant symplectic cones

We give characterizations of a finite group $G$ acting symplectically on rational surface ($\mathbb{CP}^2$ blown up at two or more points). In particular, we obtain symplectic version the dichotomy $G$-conic bundles versus $G$-del Pezzo surfaces for corresponding $G$-rational surfaces, analogous to classical result in algebraic geometry. Besides (which is completely determined case $\mathbb{CP}^2 \# N \overline{\mathbb{CP}^2}, N={2,3,4}$), also investigate equivariant minimality and cone given surface.