Real K3 surfaces with non-symplectic involution and applications. II

We consider real forms of relatively minimal rational surfaces F_m. Connected components of moduli of real non-singular curves in |-2K_{F_m}| had been classified recently for m=0, 1, 4 in math.AG/0312396. Applying similar methods, here we fill the gap for m=2 and m=3 to complete similar classification for any 0\le m\le 4 when |-2K_{F_m}| is reduced. The case of F_2 is especially remarkable and classical (quadratic cone in P^3). As an application, we finished classification of connected components of moduli of real hyper-elliptically polarized K3 surfaces and their deformations to real polarized K3 surfaces started in math.AG/0312396, math.AG/0507197. This could be important in some questions because real hyper-elliptically polarized K3 surfaces can be constructed explicitly.