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In this paper, from the group-theoretic point of view it is investigated suchclass 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 $u=u(t,x)$, $u_t=\frac{\partial u}{\partial t}$,$u_x=\frac{\partial u}{\partial x}$, $u_{xx}=\frac{\partial^2 u}{\partialx^2}$; $f(u)$ is an arbitrary smooth function of the variable $u$. Using theLie--Ovsiannikov algorithm, the group classification of the class under studyis carried out. It is shown that the kernel algebra of the full groups of theGKEs is the one-dimensional Lie algebra $\mathfrak{g}^\cap=\langle \partial_t\rangle$. Using the direct method, the equivalence group $G^\sim$ of the classis found. It is obtained six non-equivalent (up to the equivalencetransformations from the group $G^\sim$) GKEs that allow wider invariancealgebras than $\mathfrak{g}^\cap$. It is shown that, among the non-linearequations from the class, the GKE with the function $f(u)=u^{\frac43}$ has themaximal symmetry properties, namely, it admits a three-dimensional maximal Lieinvariance algebra. Using the obtained operators, it is found all possiblenon-equivalent group-invariant exact solutions of the GKE under consider.

Lie group classification and invariant exact solutions of the generalized Kompaneets equations