$\newcommand{Vol}{\text{Vol}}$

This is a cross-post.

Let $0<b<1$ be a fixed parameter, and let $(R(\theta),\theta)$ be the polar coordinates of the ellipse $$E=\{(x,y) \in \mathbb R^2 \, | \, (\frac{x}{\frac{1}{b}})^2+(\frac{y}{b})^2=1\}. \, \, \, \text{ Set}$$ $$D=\{(x,y) \in \mathbb R^2 \, | \, 0<(\frac{x}{\frac{1}{b}})^2+(\frac{y}{b})^2 \le 1\}=\cup_{0<r\le1}r E,$$ and let $f:D \to D$ be given in polar coordinates by

$$ \big(\tilde r R(\theta),\theta\big )\mapsto \bigg(\psi(\tilde r)R\big(\theta+h(\tilde r)\big),\theta+h(\tilde r)\bigg). \tag{1}$$

$f(\tilde rE)=\psi(\tilde r)E$; the "scale" is given by $\psi$ and the "phase" is given by $h$. If I am not mistaken, the Jacobian of $f$ is given by $$ Jf=\frac{\psi(\tilde r)\psi'(\tilde r)}{R(\theta)}R\big(\theta+h(\tilde r)\big)\bigg(\frac{R\big(\theta+h(\tilde r)\big)}{\tilde r R(\theta)}+\frac{R'(\theta)}{R^2(\theta)}h'(\tilde r) \bigg). $$

Question: Do there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{R}$ that satisfy $Jf=1$ everywhere, besides $\psi(x)=x$ and $h=0,\pi$ (which correspond to $f=\text{Id}$ and to $f(x,y)=-(x,y)$).

If $h$ is constant, then it easily follows that $h=0$ or $h=\pi$ and $\psi(x)=x$, so I am looking for solutions with non-constant $h$.

Comment: $$R(\theta):=\frac{b}{\sqrt{1-\big(e \cos(\theta)\big)^2}}, e=\sqrt{1-b^4}.$$