  We study a class of continuous deformations of branched complex projective
structures on closed surfaces of genus $g\geq 2$, which preserve the holonomy
representation of the structure and the order of the branch points. In the case
of non-elementary holonomy we show that when the underlying complex structure
is infinitesimally preserved the branch points are necessarily arranged on a
canonical divisor, and we establish a partial converse for hyperelliptic
structures.
