An integrable discrete-time Lagrangian system on the group of area-preserving plane diffeomorphisms \mathit{SDiff}(\mathbb{R}^{2}) is considered. It is shown that non-trivial dynamics exists only for special initial data and the corresponding mapping can be interpreted as a Bäcklund transformation for the (simple) Monge–Ampère equation. In the continuous limit, this gives the isobaric (constant pressure) solutions of the Euler equations for an ideal fluid in two dimensions. In the Appendix written by E. V. Ferapontov and A. P. Veselov, it is shown how the discrete system can be linearized using the transformation of the simple Monge–Ampère equation going back to Goursat.