Integrability and Parametric Bifurcation of the Logarithmic Emden-Fowler Equation

This study investigates the generalized Emden-Fowler equation with logarithmic nonlinearity within the framework of complete integrability by means of the S-linearizability approach. Analytical proofs show that, for the fixed nonzero logarithmic source term, the quadratic-damping coefficient is the unique value within the family a₂=k/θ compatible with the integrability condition, and that, once this damping coefficient is fixed, every source term compatible with the integrability condition necessarily takes the form θ[χ(t)+Ψ(t)lnθ], a family that contains source terms linear in θ as well as the logarithmic model investigated here. Under the parameter specialization that places the system in the scale-invariant regime, two functionally independent first integrals are constructed. These integrals are shown to generate equivalent closed-form general solutions through both a local point transformation and a generalized Sundman transformation, a result confirmed through symbolic substitution, internal consistency analysis, and independent numerical integration. The closed-form solution displays three distinct asymptotic regimes, determined by the sign of the discriminant of the characteristic equation. In particular, in contrast to the movable singularities observed in exponentially nonlinear equation classes, this family is shown to have no movable singularities; its only possible finite singular point is fixed at the origin. The S-linearizability framework provides a constructive coefficient-based route to the first integrals and the linearizing pair, consistent with the direct substitution u=lnθ. The conductivity relation provides structural motivation for possible heat-transfer applications, for which a physical interpretation requires a specified model.