This question is a special case of this one.
Let $s(\theta)>0, b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$.
Do there exist a diffeomorphism $\phi:\mathbb{S}^1 \to \mathbb{S}^1$ and a non-constant function $g:(0,1) \to \mathbb{R}$ such that $$ \phi'(\theta)=\frac{s(\theta)}{s(\phi(\theta)+g(r))}, \tag{1} $$ and $$ \big(\phi'(\theta)\big)^2+ 1/\big(\phi'(\theta)\big)^2+\big(rg'(r)+b \big( \phi(\theta)+g(r) \big)-b(\theta)\cdot \phi'(\theta)\big)^2/s^2(\theta)=C>2 \tag{2} $$ is independent of $r,\theta$?
$\phi=\text{Id}$ does not satisfy this. Equation $(1)$ reduces to $$ 1=\frac{s(\theta)}{s(\theta+g(r))}, \tag{1'} $$ and Equation $(2)$ reduces to $$ rg'(r)+b \big( \theta+g(r) \big)-b(\theta)=\sqrt{C-2}s(\theta). \tag{2'} $$ Integrating over $\theta$ and using the periodicity of $b$ (as a function over $\mathbb{S}^1$), we get $$ 2\pi rg'(r)=\sqrt{C-2}\int_{\mathbb{S}^1}s(\theta) \tag{3}, $$ so $g'(r)=\frac{c}{r}$, and $g(r)=c \log r+c_0$.
Plugging this into Equation $(1')$ we deduce that $s$ is a constant function, contradicting the assumption.
