Centering conditions for planar septic systems

We find centering conditions for the following $O$-symmetric system of degree 7: $$displaylines{ dot x=y+ x (H_2 (x,y)+H_6 (x,y)),cr dot y=-x+ y (H_2 (x,y)+H_6 (x,y)), }$$ where $H_2 (x,y)$ and $H_6 (x,y)$ are homogeneous polynomials of degrees 2 and 6, respectively. In some cases, we can find commuting systems and first integrals for the original system. We also study the geometry of the central region.