  We study, in dimensions $N\geq 3$, the family of first integrals of an
incompressible flow: these are $H^{1}_{loc}$ functions whose level surfaces are
tangent to the streamlines of the advective incompressible field. One main
motivation for this study comes from earlier results proving that the existence
of nontrivial first integrals of an incompressible flow $q$ is the main key
that leads to a "linear speed up" by a large advection of pulsating traveling
fronts solving a reaction-advection-diffusion equation in a periodic
heterogeneous framework. The family of first integrals is not well understood
in dimensions $N\geq3$ due to the randomness of the trajectories of $q$ and
this is in contrast with the case N=2. By looking at the domain of propagation
as a union of different components produced by the advective field, we provide
more information about first integrals and we give a class of incompressible
flows which exhibit `ergodic components' of positive Lebesgue measure (hence
are not shear flows) and which, under certain sharp geometric conditions, speed
up the KPP fronts linearly with respect to the large amplitude. In the proofs,
we establish a link between incompressibility, ergodicity, first integrals, and
the dimension to give a sharp condition about the asymptotic behavior of the
minimal KPP speed in terms the configuration of ergodic components.
