We study the processes e+e− → 2ðπþπ−Þπ0π0π0γ and 2ðπþπ−Þπ0π0ηγ in which an energetic photon is radiated from the initial state. The data were collected with the BABAR detector at SLAC. About 14 000 and 4700 events, respectively, are selected from a data sample corresponding to an integrated luminosity of 469 fb−1. The invariant mass of the hadronic final state defines the effective eþe− center-of-mass energy. The center-of-mass energies range from threshold to 4.5 GeV. From the mass spectra, the first ever measurements of the eþe− → 2ðπþπ−Þπ0π0π0 and the eþe− → 2ðπþπ−Þπ0π0η cross sections are performed. The contributions from ωπþπ−π0π0, η2ðπþπ−Þ, and other intermediate states are presented. We observe the J=ψ and ψð2SÞ in most of these final states and measure the corresponding branching fractions, many of them for the first time.
