Homogeneous fluxes, branes and a maximally supersymmetric solution of M-theory

We find M-theory solutions with homogeneous fluxes for which the spacetime is a lorentzian symmetric space. We show that generic solutions preserve sixteen supersymmetries and that there are two special points in their moduli space of parameters which preserve all thirty-two supersymmetries. We calculate the symmetry superalgebra of all these solutions. We then construct various M-theory and string theory branes with homogeneous fluxes and we also find new homogeneous flux-brane solutions.