  A nonzero locally nilpotent linear derivation of the polynomial algebra K[X]
in d variables over a field K of characteristic 0 is called a Weitzenboeck
derivation. The classical theorem of Weitzenboeck states that the algebra of
constants (which coincides with the algebra of invariants of a single unipotent
transformation) is finitely generated. Similarly one may consider the algebra
of constants of a locally nilpotent linear derivation of a finitely generated
(not necessarily commutative or associative) algebra which is relatively free
in a variety of algebras over K. Now the algebra of constants is usually not
finitely generated. Except for some trivial cases this holds for the algebra of
constants of the free metabelian Lie algebra L/L" with d generators. We show
that the vector space of the constants in the commutator ideal L'/L" is a
finitely generated module over the algebra of constants in K[X]. For small d,
we calculate the Hilbert series of the algebra of constants in L/L" and find
the generators of the module of the constants in L'/L". This gives also an
(infinite) set of generators of the Lie algebra of constants in L/L".
