  The paper gives the complete characterization of all graded nilpotent Lie
algebras with infinite-dimensional Tanaka prolongation as extensions of graded
nilpotent Lie algebras of lower dimension by means of a commutative ideal. We
introduce a notion of weak characteristics of a vector distribution and prove
that if a bracket-generating distribution of constant type does not have
non-zero complex weak characteristics, then its symmetry algebra is necessarily
finite-dimensional. The paper also contains a number of illustrative algebraic
and geometric examples including the proof that any metabelian Lie algebra with
a 2-dimensional center always has an infinite-dimensional Tanaka prolongation.
