  Motivated by the framework constructed by Brugnano and Casulli $[$SIAM J.
Sci. Comput. 30: 463--472, 2008$]$, we analyze the finite termination property
of the generalized Netwon method (GNM) for solving the absolute value equation
(AVE). More precisely, for some special matrices, GNM is terminated in at most
$2n + 2$ iterations. A new result for the unique solvability and unsolvability
of the AVE is obtained. Numerical experiments are given to demonstrate the
theoretical analysis.
