  We study one-dimensional itinerant ferromagnets with Heisenberg symmetry near
a ferromagnetic quantum critical point. It is shown that the Berry phase term
arises in the effective action of itinerant ferromagnets when the full SU(2)
symmetry is present. We explicitly demonstrate that dynamical critical exponent
of the theory with the Berry term is $z=2 +{\rm O}(\epsilon^2)$ in the sense of
$\epsilon$ expansion, as previously discovered in the Ising limit. It appears,
however, that the universality class at the interacting fixed point is not the
same. We point out that even though the critical theory in the Ising limit can
be obtained by the standard Hertz-Millis approach, the Heisenberg limit is
expected to be different. We also calculate the exact electron Green functions
$G(x,t=0)$ and $G(x=0,t)$ near the transition in a range of temperature, which
can be used for experimental signatures of the associated critical points.
