  Recently, the $\mu$-deformation-based approach to modeling dark matter, which
exploits $\mu$-deformed thermodynamics, was extended to the study of galaxy
halo density profile and of the rotation curves of a number of (dwarf or low
brightness) galaxies. For that goal, $\mu$-deformed analogs of the Lane--Emden
equation (LEE) have been proposed, and their solutions describing density
profiles obtained. There are two seemingly different versions of $\mu$-deformed
LEE which possess the same solution, and so we deal with their equivalence.
From the latter property we derive new, rather unusual, $\mu$-deformed
Heisenberg algebra (HA) for the position and momentum operators, and present
the $\mu$-HA in few possible forms (each one at $\mu\to0$ recovers usual HA).
The generalized uncertainty relation linked with the new $\mu$-HA is studied,
along with its interesting implications including the appearance of the
quadruple of both maximal and minimal lengths and momenta.
