  I consider the partition function of the inhomogeneous 6-vertex model defined
on the $n$ by $n$ square lattice. This function depends on 2n spectral
parameters $x_i$ and $y_i$ attached to the horizontal and vertical lines
respectively. In the case of domain wall boundary conditions it is given by
Izergin-Korepin determinant. For $q$ being a root of unity the partition
function satisfies to a special linear functional equation. This equation is
particularly good when the crossing parameter $\eta=2\pi/3$. In this case it
can be used for solving some of the problems related to the enumeration of
alternating sign matrices. In particular, it is possible to reproduce the
refined ASM distribution discovered by Mills, Robbins and Rumsey and proved by
Zeilberger. Further, it is well known that the partition function is symmetric
in the $\{x\}$ and as well in the $\{y\}$ variables. I have found that in the
case of $\eta=2\pi/3$, the partition function is symmetric in the union $\{x\}
\cup \{y\}$! This nice symmetry is used to find some relations between the
numbers of such alternating sign matrices of order $n$ whose two '1' are
located in fixed positions on the boundary of the matrices. Finally I derive
the equation giving `top-bottom double refined' ASM distribution.
