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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.
