Presentations of free nilpotent metabelian lie algebras

F, be a free Lie algebra of rank n. For m,n≥2, we have investigated the presentations of the free metabelian nilpotent Lie algebra M_(n,m)≅F⁄(γ_m (F)+F^'' ). We have shown that for 〖m=2,3,4 ,M〗_(n,m) admits a presentation whose set of defining relations is the set H_m where H_m is the set of basic monomials of length m. The same result is obtained for M_2,5. For m>5 and n=2 we have found a presentation of M_(2,m). For n≥2 ve m≥5 we present a presentation of M_(n,m) and we refine this presentation to a better one. Finally we refine the presentation of M_(n,m) and we obtained sharper presentations of M_(n,5) and M_3,7.