  In this paper, from the group-theoretic point of view it is investigated such
class of the generalized Kompaneets equations (GKEs):
$$u_t=\frac1{x^2}\cdot\left[x^4(u_x+f(u))\right]_x, \ (t,x) \in \mathbb{R}_{+}
\times \mathbb{R}_{+},$$ where $u=u(t,x)$, $u_t=\frac{\partial u}{\partial t}$,
$u_x=\frac{\partial u}{\partial x}$, $u_{xx}=\frac{\partial^2 u}{\partial
x^2}$; $f(u)$ is an arbitrary smooth function of the variable $u$. Using the
Lie--Ovsiannikov algorithm, the group classification of the class under study
is carried out. It is shown that the kernel algebra of the full groups of the
GKEs is the one-dimensional Lie algebra $\mathfrak{g}^\cap=\langle \partial_t
\rangle$. Using the direct method, the equivalence group $G^\sim$ of the class
is found. It is obtained six non-equivalent (up to the equivalence
transformations from the group $G^\sim$) GKEs that allow wider invariance
algebras than $\mathfrak{g}^\cap$. It is shown that, among the non-linear
equations from the class, the GKE with the function $f(u)=u^{\frac43}$ has the
maximal symmetry properties, namely, it admits a three-dimensional maximal Lie
invariance algebra. Using the obtained operators, it is found all possible
non-equivalent group-invariant exact solutions of the GKE under consider.
