On $(\lambda ,\mu )$-classes on the Engel group

The purpose of this note is to compare the properties of the symbolic pseudo-differential calculus on the Heisenberg and on the Engel groups; nilpotent Lie groups of 2-step and 3-step, respectively. Here we provide a preliminary analysis of the structure and of the symbolic calculus with symbols parametrized by $(\lambda ,\mu )$ on the Engel group, while for the case of the Heisenberg group we recall the analogous results on the $\lambda$-classes of symbols.