Generalized Buchdahl equations as Lie–Hamilton systems from the "book" and oscillator algebras: quantum deformations and their general solution

We revisit the nonlinear second-order differential equations $\ddot{x}(t)=a(x)\dot{x}(t{)}^2+b(t)\dot{x}(t),$ where $a(x)$ and $b(t)$ are arbitrary functions on their argument from the perspective of Lie–Hamilton systems. For the particular choice $a(x)=3/x$ and $b(t)=1/t$, these equations reduce to the Buchdahl equation considered in the context of general relativity. It is shown that these equations are associated with the "book" Lie algebra ${\mathfrak{b}}_2$ by determining a Lie–Hamilton system for which the corresponding $t$-dependent Hamiltonian and the general solution of the equations are given. The procedure is illustrated considering several particular cases. We also make use of the quantum deformation of ${\mathfrak{b}}_2$ with the quantum deformation parameter $z$ (where $q={\mathrm{e}}^z$), leading to a deformed generalized Buchdahl equation. Applying the formalism of Poisson–Hopf deformations of Lie–Hamilton systems, we derive the corresponding deformed $t$-dependent Hamiltonian, as well as its general solution. The generalized Buchdahl equation is further extended to the oscillator Lie–Hamilton algebra ${\mathfrak{h}}_4\supset {\mathfrak{b}}_2$, together with its quantum deformation, and the corresponding (deformed) equations are also analyzed for their exact solutions. The presence of the quantum deformation parameter $z$ is interpreted as the introduction of an integrable perturbation of the (initial) generalized Buchdahl equation, which is described in detail in its linear approximation. Finally, it is also shown that, under quantum deformations, the higher-dimensional deformed generalized Buchdahl equations from either the book or the oscillator algebras do not reduce to a sum of copies of the initial system but to intrinsic coupled systems governed by $z$.