Could there be exact solutions to the Lane-Emden equation for real n≥0 other than 0, 1, or 5? This Astronomy SE answer says

With a constant $k$ and the polytrop index $n$. This is a result of the solutions of the Lane-Emden equation
$$\frac{1}{\xi^2} \frac{\mathrm{d}}{\mathrm{d}\xi} \left(\xi^2\frac{\mathrm{d}\theta}{\mathrm{d}\xi}\right) + \theta^n = 0$$
which is a dimensionless form of the Poisson equation for a radially-symmetric self-gravitating polytropic fluid, thus where density follows a function of the form $\varrho = \varrho_c \theta^n$ with a central density $\varrho_c$.
This equation can be solved exactly for polytrop index 0 (isobaric polytrope), 1 (isothermal polytrope) and 5 (limited use as it results in infinite stellar radius)...

Question: Wikipedia notes that exact solutions (not depending on converging series) are known only for $n=(0, 1, 5)$. Could there be as-yet undiscovered exact solutions for other real $n \ge 0$? Or has it been shown that there are no others possible?
Potentially related:

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*I have a special solution for the Lane-Emden equation. Can I use it to find the general solution?
 A: "Analytical solving is possible" means that the solution can be expressed with a finite number of standard functions (elementary and special functions). But this doesn't mean that analytical solving is impossible for ever. If new convenient special functions were defined and standardized, the solving would become possible thanks to those functions.
This is the same as the Bessel ODEs for example. They were not anatically solvable and become solvable when some new special functions named "Bessel functions" where defined and standardized.
Citation : "A special function has to acquire a background of property, descriptions,
formulas and derivations as extended as possible. So, it will be preferable to simply refer to a particular part of the background, instead of searching and redoing a development by ourselves. Before becoming a referenced special function, its name has to be spread in the literature in order to become familiar. More importantly, the function should be useful in a branch of mathematics or physics." (Citation from p.3 in https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function ).
Generally, a function has a long way to go before acquiring the honorific status of special
function. Probably a long way too for the expected special fonctions related to the Lane-Emden equation.
