We introduce and study the class Fₗambda of autonomous ordinary differential equations du/dt = u exp (lambda u), parametrized by lambda in R. This family is shown to be the unique family of the form du/dt = g (u) Phi (lambda u) whose associated vector space A₆, ₇₈ = Vect g (u) ᵏ Phi (lambda u): k >= 1 is stable under both natural derivation operators d/dlambda and d/du — a purely structural constraint that forces Phi (s) = exp (s) and g (u) = u uniquely (up to normalization). The explicit linearizer of Fₗambda is identified as the exponential integral E1, and the full hierarchy Aₙ: du/dt = uⁿ exp (lambda u) is shown to be linearized by lambda^n-1 Eₙ (lambda u). The bifurcation structure at lambda = 0 is completely classified across the hierarchy, revealing three distinct asymptotic regimes. A complete classification of stable pairs (g, Phi) establishes that Fₗambda is the unique class with a transcendental non-elementary linearizer. The operators stabilizing Aₔ, ₄ₗ (ₒ) are shown to generate a three-dimensional solvable Lie algebra. Complexification of the parameter lambda = i mu reveals a hidden exact first integral Omega = pi/2 - Si (mu u₀), invisible from the real dynamics, which combines a continuous analytical part with a discrete correction 2pi K (t) in 2pi Z counting branch cut crossings. This conservation law is preserved exactly along complex trajectories as a direct consequence of the linearization structure. The algebraic structure of Fₗambda is shown to force a canonical PDE analogue: the class Pₗambda^+, obtained by requiring that W = E1 (lambda U) satisfies the heat equation, inherits the full hierarchy, the solvable Lie algebra, and the topological invariant as a spatially-indexed conservation law field. The complexification lambda = i mu is extended to the full hierarchy Aₙ: the exact first integrals Omega^ (n) are computed explicitly for n = 2, 3, 4 via the recurrence on Eₙ, revealing that the topological invariant pi/2 - Si (mu u₀) persists at every level, weighted by (mu u₀) ^n-1. A complete classification of all equations of the form dₜ U + f (U) dₓ U = nu dₗₗU - (lambda + 1/U) (dₓ U) ² + h (U, t) that are exactly linearized by W = E1 (lambda U) is established: they are precisely those with f (U) = F (E1 (lambda U) ) and h (U, t) = -U exp (lambda U) H (E1 (lambda U), t) for arbitrary functions F, H. Numerical validation of the predicted extinction rates exp (-t) and exp (-3t) for the PDE Pₗambda^+ is provided, confirming the theory. Version
Judicael Brindel (Thu,) studied this question.