  It is introduced an analogue of the orbit-breaking subalgebra for the case of
free flows on locally compact metric spaces, which has a natural approximate
structure in terms of a fixed point and any nested sequence of central slices
around this point. It is shown that in the case of minimal flows admitting a
compact Cantor central slice, the resulting $C^*$-algebra is the stabilization
of the Putnam orbit-breaking subalgebra associated to the induced homeomorphism
on the central slice.
