The F lambda Class: Algebraic Stability, Explicit Linearization, and Topological Invariants of a Canonical Family of ODEs

We introduce and study the class F_lambda 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 Ag,Phi = 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 F_lambda is identified as the exponential integral E1, and the full hierarchy A_n : du/dt = u^n exp(lambda u) is shown to be linearized by lambdaⁿ⁻¹ E_n(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 F_lambda is the unique class with a transcendental non-elementary linearizer. The operators stabilizing Au,exp(s) 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 u_0), 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 F_lambda is shown to force a canonical PDE analogue: the class P_lambda^+, 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 A_n: the exact first integrals Omega^(n) are computed explicitly for n = 2, 3, 4 via the recurrence on E_n, revealing that the topological invariant pi/2 - Si(mu u_0) persists at every level, weighted by (mu u_0)ⁿ⁻¹. A complete classification of all equations of the form d_t U + f(U) d_x U = nu [ dₓₓU - (lambda + 1/U) (d_x 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 P_lambda^+ is provided, confirming the theory. Version