Shift-symmetric

A Poincar\`{e} invariant, local scalar field theory in which the Lagrangian and equation of motion contain only up to second-order derivatives fields is called generalized Galileon. The covariant version it four dimensions Horndeski theory, has been vigorously studied applications inflation dark energy. In this paper, we study a class multi-field extensions Galileon theory. By imposing shift $SO(N)$ symmetries on all currently known multi-Galileon terms general dimensions, find that structure uniquely determined parameterized by series coupling constants. We also tensor perturbation shift-symmetric $SO(3)$ dimensions. perturbations can obtain mass term stemming from same symmetry breaking pattern as solid inflation. gives rise new cubic interactions modes, suggesting existence type primordial non-Gaussianity.