Maximal Poisson Commutative subalgebras for truncated parabolicsubalgebras of maximal index in sln

We show that several properties of the semisimple algebras carry over to a certain family of parabolic subalgebras of maximal index in sl_n. More precisely we prove an analogue of Kostant's slice theorem [B. Kostant, Amer. J. Math. 85 (1963), 327-404] for these algebras and construct a maximal Poisson commutative subalgebra in the symmetric algebra, following the theory presented in [A.S. Mishchenko and A.T. Fomenko, Math. USSR-Izv. 12 (1978), 371-389]. These results are quite remarkable since these algebras do not admit appropriate sl₂-triples.