Let begin our discussion with the following curve $$ x=ye^{-y}.$$ Let choose a parametrization variable $z$ and we parametrize the above curve $$ x(z):=ze^{-z}, y(z)=z $$ We find the ramification point of the curve that is when $\frac{\partial}{\partial x }x(z)=0$ that is when $z=1$. We are interested in the involution around the ramification point. That is the function $s(z)$ such that $x(s(z)) = x(z)$. So we did a shift of coordinate by 1 that is we move the ramification point at $0$. Hence the new parametric equation is $$x(z):=(1+z)e^{-(1+z)}, y(z)=1+z $$ Now to solve for the involution I put in maple $$x(z)=x(s) $$ and ask to solve in $s$ in terms of z it gives me the following function $$s(z)= -\text{LambertW}\Big(-(1+z)e^{(-1-z)} \Big)$$ when I take the expansion at $z=0$ of s(z) it give the following series
$$s(z):= -z+2/3\,{z}^{2}-4/9\,{z}^{3}+{\frac {44\,{z}^{4}}{135}}-{\frac {104\,{ z}^{5}}{405}}+{\frac {40\,{z}^{6}}{189}}-{\frac {7648\,{z}^{7}}{42525} }+{\frac {2848\,{z}^{8}}{18225}}-{\frac {31712\,{z}^{9}}{229635}} .$$ which is the invloution at the ramification point $z=0$.
With the success of the above calculation, we want to find involution for the parametrised curve
$$x(z):= z{{\rm e}^{-w_{{1}} \left( q_{{2}}{z}^{2}+q_{{1}}z \right) }} y(z):=q_{{2}}{z}^{2}+q_{{1}}z. $$
If we put $q_2 =0,\ q_1 =1$ we get back our curve we started with. The ramification point for the above curve occur at the point
$$z=1/4\,{\frac {-q_{{1}}w_{{1}}+\sqrt {{q_{{1}}}^{2}{w_{{1}}}^{2}+8\,q_{{ 2}}w_{{1}}}}{q_{{2}}w_{{1}}}} $$ and $$z=-1/4\,{\frac {q_{{1}}w_{{1}}+\sqrt {{q_{{1}}}^{2}{w_{{1}}}^{2}+8\,q_{{ 2}}w_{{1}}}}{q_{{2}}w_{{1}}}} $$ I put in maple but could not find the involution around the ramification point. Is there any way I can find a series approximation of the ramification point. Also, it should satisfy that fact that if I substitute $q_2 =0, q_1 =1$ then it should give back the original ramification.
I am not an expert in LambertW function if there is a way to see the involution as some combination of LambertW function please let me know.