A new dimension-reduction method, the Invariant Constrained-equilibrium Edge Pre-Image Curve (ICE-PIC) method, to simplify chemical kinetics has recently been developed by Ren et al. [Z. Ren, S.B. Pope, A. Vladimirsky, J.M. Guckenheimer, J. Chem. Phys. 124 (2006) 114111]. In the present work, the ICE-PIC method is first applied to the homogeneous autoignition of stoichiometric methane/air and its accuracy is shown to compare favorably to those of other methods (QSSA and RCCE). For inhomogeneous systems such as flames, spatial transport by molecular diffusion causes a small perturbation of the composition away from the attracting, low-dimensional, invariant manifold identified by the ICE-PIC method. A “close-parallel” assumption is introduced which allows this perturbation to be determined, and leads to an additional “transport coupling” term in the evolution equation for the reduced variables. For the test case of a steady, one-dimensional, laminar, methane/air flame, it is shown that the inclusion of transport coupling can reduce the dimension-reduction errors by a factor of 100. The ICE-PIC method with eight degrees of freedom (including transport coupling) exhibits comparable accuracy to a quasi-steady state assumption (QSSA) reduced mechanism with 12 degrees of freedom.