  We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions
as a function of an applied magnetic field $h$. At the Uimin-Lai-Sutherland
(ULS) critical point we find a gapless to gapless transition revealed by the
dynamical structure factor $S(q,\omega)$ as a function of $h$. At $h=0$, the
envelope of the lowest energy excitations goes soft at {\it two points}
$q_1=2\pi/3$ and $q_2=4\pi/3$, dubbed the phase A. With increasing field, the
spectral peaks at each of the gapless points bifurcate, making in total {\it
four} soft modes, and combine to form a new set of excitations that soften at a
{\it single} point $q=\pi$ at $h_{c1}\approx 0.94$. Beyond $h_{c1}$ the system
enters another gapless B-phase until the transition at $h_{c2}=4$ to the fully
polarized phase. We compare the ULS model results with those for the AKLT model
as a representative of gapped Haldane phase. We explain the mechanism of the
gapless to gapless transition in the ULS model using its conserved charges and
a spinon band picture. We also discuss the universality of central charges of
the BLBQ family of models subjected to a magnetic field.
