Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ are taken to be real quantities with $a = b$ considered separately as limiting cases and $m$, an integer.)
For $m =1$, the reflected curve is
$$x = \frac{y}{(1-ay)(1-by)},$$
so solving the quadratic
$$(1-ay)(1-by)x - y = (a b x) y^2 - (1+ (a + b)x) y + x = 0$$
for $y$ in terms of $x$ gives the two branches for the reflected curve
$$y = f_{1,\pm}^{<-1>}(x) = \frac{1+(a+b)x \pm \sqrt{(1+(a+b)x)^2 - 4abx^2}}{2abx}.$$
Expansion of the reflected curve through the origin, $y=f_{1,-}^{<-1>}(x)$, as a power series gives a bivariate version of the Narayana polynomials of OEIS A001263, i.e., the h-polynomials of the associahedra, in the denominators wheres expansion of the section of $y = f_1(x)$ through the origin gives the bivariate complete homogeneous symmetric polynomials $h_n(a,b) = \frac{a^{n+1} - b^{n+1}}{a-b}$. Changing the bivariate variables relates the Narayana polynomials to numerous other series of polynomials with combinatorial models significant in geometry/topology and analysis, such as the f-polynomials of the associahedra. Expansions as Laurent series away from the origin and about $x= \infty$ lead to the Narayana polynomials and $h_n(\frac{1}{a}, \frac{1}{b})$ as well, associated with the free cumulants and moments of free probability theory. For some combinatorics and physics related to the bivariate complete homogeneous symmetric polynomials, see this MO-Q.
A few researchers and I have explored aspects of these power and Laurent expansions for $m$ any integer. The focus of this question is on another type of expansion over different sections of the curve--Puiseux-type expansions that cover sections of the reflected curve that the power and Laurent series expansions about the origin and infinity do not.
I've explored this for $m = 1$. For example, for $a = 3$ and $b=7$ (see DESMOS link below), the upper section of the reflected curve / compositional inverse in the first quadrant between $x=|c_{2}|=|\ \frac{-\left(a+b\right)+2\sqrt{ab}}{\left(a-b\right)^{2}}| $ and $x = |c_{1}|=|\ \frac{-\left(a+b\right)-2\sqrt{ab}}{\left(a-b\right)^{2}}|$ is given by
$$S\left(x\right)=\frac{1+\left(a+b\right)x-p\left(x\right)}{2abx}$$
with
$$p\left(x\right)=\left(a-b\right)\sqrt{c_{1}c_{2}}\sqrt{\frac{-x}{c_{2}}}\left(\sum_{m=-\infty}^{\infty}\left(\frac{-x}{c_{1}}\right)^{m}\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}\right)!}{\left(m+k\right)!\left(\frac{1}{2}-\left(m+k\right)\right)!}\frac{\left(\frac{1}{2}\right)!}{k!\left(\frac{1}{2}-k\right)!}\left(\frac{c_{2}}{c_{1}}\right)^{k}\right) .$$
The parameters $c_1$ and $c_2$ are determined from the zeros of the expression in the square root of $f_{1,\pm}^{<-1>}(x)$.
The associated Laurent series expansion for the corresponding section of the curve $y= f_{1,+}(x)$ is
$$f_{1,+}(x) = \sum_{n=1}^{\infty}\left(\frac{b^{\left(-n\right)}-a^{\left(-n\right)}}{a-b}\right)\ x^{\left(-n\right)} ,$$
involving $h_k(\frac{1}{a},\frac{1}{b})$.
(This is a DESMOS graph of the related curves and parameters with truncated series. DESMOS knows how to handle $1/(-n)!$ as vanishing for $n = 1,2,3, \cdots$ and can plot $x=f(y)$ just as well as $y =f(x)$.)
The coefficients of the Narayana polynomials are a finite refinement / splitting of the Catalan numbers (OEIS A000108, check with $a=b=1$) whereas the numerical coefficients of the sum in $p(x)$ for $c_1=c_2 =1$ are $\frac{1}{(m+\frac{1}{2})!}$ (here $m$ is just the dummy summation index) containing the reciprocal Catalan numbers (or odd double factorials) for $m > 0$ and the Catalan numbers for $m <0$ and the square root of pi, powers of $2$, and the factorials.
Determining the Puiseux series expansions for $f_{m}^{<-1>}(x)$ for $m = 2$ involves solving a cubic equation; $m=3$, a quartic; and so on, becoming exceedingly more difficult to explore. (The generalized Fuss-Catalan, or Fuss-Narayana, number sequences flagged by integer $m$ and refinements of them appear.)
My question is
Is there any literature that has explored such Puiseux-type expansions for sections of the reflected curve for any $|m| \geq 1$?
