  We give construction of singular K3 surfaces with discriminant 3 and 4 as
double coverings over the projective plane. Focusing on the similarities in
their branching loci, we can generalize this construction, and obtain a three
dimensional moduli space of certain K3 surfaces which admit infinite
automorphism groups. Moreover, we show that these K3 surfaces are characterized
in terms of the configuration of the singular fibres and a global section, and
also in terms of periods.
