One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial derivatives, connected with volume-preserving diffeomorphisms. In this paper, we determine the structure space of the divergence-free Lie algebras associated with pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra.  2001 Academic Press 1. INTRODUCTION Volume-preserving diffeomorphisms are important transformations on manifolds. Lie groups of volume-preserving diffeomorphisms are fundamental structures in the theory of idealized fluids. They have been studied through their Lie algebras, whose elements are divergence-free. One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial derivatives. Graded generalizations of the Lie algebras of Special type have been 557