Recent papers have shown that
${{C}^{1}}$
maps
$F:\,{{\mathbb{R}}^{2}}\,\to {{\mathbb{R}}^{2}}$
whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or
$F$
is a polynomial. Specifically,
$F\,=\,(u,\,v)$
must take the form
$$u\,=\,ax\,+\,by\,+\,\beta \phi (\alpha x\,+\,\beta y)\,+\,e$$
$$v\,=\,cx\,+\,dy\,-\,\alpha \phi \,(\alpha x\,+\,\beta y)\,+\,f$$
for some constants
$a,\,b,\,c,\,d,\,e,\,f,\,\alpha ,\,\beta $
and a
${{C}^{1}}$
function
$\phi $
in one variable. If, in addition, the function
$\phi $
is not affine, then
1
$$\alpha \beta (d\,-\,a)\,+\,b{{\alpha }^{2}}\,-\,c{{\beta }^{2}}\,=\,0.$$
This paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are
$\pm 1/2$
and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge—Ampère equation.