A condition for an area-preserving mapping to be in the Engel's form

We establish explicit expressions of restrictions on the coefficients of no nlinear terms in a two-dimensional area-preserving polynomial map imposed b y the property of area preserving. We also establish a necessary and suffic ient condition for a two-dimensional area-preserving generic polynomial map to be in the Engel's form. The condition is that coefficients of nonlinear terms in the first mapping equation are proportional to the corresponding ones in the second mapping equation. As an application of the results, we d iscuss how to regain the area-preserving property lost in truncating a two- dimensional area-preserving map. (C) 2000 American Institute of Physics. [S 0022-2488(00)06905-X].