  We investigate the filtration corresponding to the degree function induced by
a non-zero locally nilpotent derivation and its associated graded algebra. We
show that this kind of filtration, referred to as the LND-filtration, is the
ideal candidate to study the structure of semi-rigid k-domains, that is,
k-domains for which every non-zero locally nilpotent derivation gives rise to
the same filtration. Indeed, the LND-filtration gives a very precise
understanding of these structure, it is impeccable for the computation of the
Makar-Limanov invariant, and it is an efficient tool to determine their
isomorphism types and automorphism groups. Then, we construct a new interesting
class of semi-rigid k-domains in which we elaborate the fundamental requirement
of LND-filtrations. The importance of these new examples is due to the fact
that they possess a relatively big set of invariant sub-algebras, which can not
be recoverd by known invariants such as the Makar-Limanov and the Derksen
invariants. Also, we define a new family of invariant sub-algebras as a
generalization of the Derksen invariant. Finally, we introduce an algorithm to
establish explicit isomorphisms between cylinders over non-isomorphic members
of the new class, providing by that new counter-examples to the cancellation
problem.
