Electromagnetic anapoles are nonradiating oscillations of electric charge. They are known to be formed by a combination of an ordinary and a toroidal multipole. In this work, we introduce a different family of anapoles based on a Cartesian electric-current multipole expansion and propose a way to create the lowest-order, centrosymmetric member of this family. We also find a nontoroidal multipole structure that radiates as an electric dipole despite its zero total dipole moment. Radiation-free charge oscillations are interesting not only in view of their unique fundamental properties, but also in view of practical applications, e.g., in near-field-based optical sensing and detection.