Generalized orbital angular momentum symmetry in parametric amplification

We investigate interesting symmetry properties verified by the down-converted beams produced in optical parametric amplification with structured light. show that Poincar\'e sphere symmetry, previously demonstrated for first-order spatial modes, translates to a multiple structure higher orders. Each one of these spheres is associated two-dimensional subspace defined different value orbital angular momentum. Therefore, first order modes reproduced independently each subspace. This effect can be useful parallel control correlated beams.