  We consider the Izergin-Korepin determinant [1] together with another
determinant which was invented by Kuperberg [2]. He used these determinants to
prove a formula for the total number of half-turn symmetric alternating sign
matrices of even order conjectured by Robbins [3]. By developing further the
method that was described in our previous paper [4], we obtain a closed
nonlinear recurrence system for these determinants. It can be used in various
ways. For example, in this paper, we obtain formula (29) for the refined
enumeration of half-turn symmetric alternating sign matrices of even order.
