  We study the effect of a hidden gauge symmetry on complex holomorphic
systems. For this purpose, we show that intrinsically any holomorphic system
has this gauge symmetry. We establish that this symmetry is related to the
Cauchy-Riemann equations, in the sense that the associated constraint is a
first class constraint only in the case that the potential be holomorphic. As a
consequence of this gauge symmetry on the complex space, we can fix the gauge
condition in several ways and project from the complex phase-space to real
phase space. Different projections are gauge related on the complex phase-space
but are not directly related on the real physical phase-space.
