  Some cases of the LFED Conjecture, proposed by the second author [Z3], for
certain integral domains are proved. In particular, the LFED Conjecture is
completely established for the field of fractions $k(x)$ of the polynomial
algebra $k[x]$, the formal power series algebra $k[[x]]$ and the Laurent formal
power series algebra $k[[x]][x^{-1}]$, where $x=(x_1, x_2, \dots, x_n)$ denotes
$n$ commutative free variables and $k$ a field of characteristic zero.
Furthermore, the relation between the LFED Conjecture and the Duistermaat-van
der Kallen Theorem [DK] is also discussed and emphasized.
