  In this paper we present a method for extending the blowup method, in the
formulation of Krupa and Szmolyan, to flat slow manifolds that lose
hyperbolicity beyond any algebraic order. Although these manifolds have
infinite co-dimension, they do appear naturally in certain settings. For
example in (a) the regularization of piecewise smooth systems by $\tanh$, (b) a
model of aircraft landing dynamics, and finally (c) in a model of earthquake
faulting. We demonstrate the approach on a simple model system and the examples
(a) and (b).
