Parametric equation of $x^y=y^x$ curve

Suppose you take $x=e^{r\cos\theta}$ and $y=e^{r\sin\theta}$ for $\theta\in(0,\pi/2)$. Then you want to solve $$ e^{r\cos\theta e^{r\sin\theta}}=e^{r\sin\theta e^{r\cos\theta}} $$ for $r$. Taking the logarithm of both sides and dividing through by $r$ (losing the $r=0$ solution in the process) gives $$ \cos\theta e^{r\sin\theta} = \sin\theta e^{r\cos\theta}, $$ hence $$ \tan\theta = e^{r(\sin\theta - \cos\theta)}, $$ or $$ r(\sin\theta - \cos\theta) = \ln\tan\theta. $$ This is satisfied for $\theta=\pi/4$, for any $r$, which gives the line $y=x$. The piece you're interested in is given by $$ r=\frac{\ln\tan\theta}{\sin\theta - \cos\theta}. $$ Plugging this back in, you have $$ (x,y)=\left(\exp\left(\frac{\ln\tan\theta}{\tan\theta - 1}\right), \exp\left(\frac{\ln\tan\theta}{1-\cot\theta}\right) \right) \\ =\left(\left(\tan\theta\right)^{\frac{1}{\tan\theta-1}}, \left(\tan\theta\right)^{\frac{1}{1-\cot\theta}}\right), $$ or simply $$ (x,y)=\left(t^{1/(t-1)}, t^{t/(t-1)}\right) $$ for $t \in (0,\infty)$. (Yes, there is a hole at $t=1$.)

I've done this here before, but it's quicker to write it again than to try to find it.

Let $y = rx$. $x^y = y^x$ becomes $x^{rx} = (rx)^x$ or $x^r = rx$ or $x^{r-1} = r$ or $x = r^{1/(r-1)}$ and $y = rx = r^{1+1/(r-1)} =r^{r/(r-1)} $.

If $r=1$, all values of $x$ work since the condition is $x^{r-1} = r$.

Edit: I just see that you already mention this one in your question... But unlike you state there, it does include $(e,e)$ namely for $t=0$.

Basically a reparameterisation of other answers but $$ x(t) = \exp\left(\dfrac{t}{e^t-1}\right) \quad y(t)=\exp\left(\dfrac{t}{1-e^{-t}}\right) $$ for $t\in \mathbb{R}$ seems to be a nice and symmetric one. Note that these expressions extend nicely over $t=0$ and $$\left(x(0), y(0)\right) = (e,e)$$ and $$ \left(x(-t), y(-t)\right) = \left(y(t), x(t)\right).$$