We propose a new single-step mechanism for the combustion of arbitrary hydrocarbons and alcohols. Unlike most single-step models, no tabulation is required, as the method builds upon a new analytical description of the thermochemical equilibrium of fuel-oxidizer mixtures including dihydrogen and carbon monoxide-two species usually discarded in one-step descriptions-yielding correct adiabatic temperature. The single-step chemistry includes varying stoichiometric coefficients, ensuring a convergence towards thermochemical equilibrium regardless of the local state. The reaction rate is then carefully adjusted to reproduce accurately premixed flames. To tackle ignition simultaneously, an additional passive scalar advection-diffusion-reaction equation is introduced, with a rate fitted on ignition delays. The scalar then serves as an efficiency to modify the single-step reaction rate in autoignition configurations. The obtained scheme is then validated for a wide range of equivalence ratios on homogeneous reactors, premixed flames, a triple flame, and a counterflow diffusion flame. The new analytical thermochemical equilibrium formulation may also serve in speeding up infinitely fast chemistry calculations.