Jacobian conjecture, two-dimensional case

The Jacobian Conjecture was first formulated by O. Keller in 1939. In the modern form it supposes injectivity of the polynomial mapping $f$: $\mathbb{R}^n \to \mathbb{R}^n$ ($\mathbb{C}^n \to \mathbb{C}^n$) provided that jacobian $J_f\equiv \mathrm{const}\ne0$. In this note we consider structure of polynomial mappings $f$ that provide $J_f\equiv \mathrm{const} \ne0$.