In the cholesteric liquid crystal helix structure, the director, n, rotates in a plane about an axis, t(0) perpendicular to n, with a constant twist, n.curln = -q. The inverse helix pitch is defined by q = 2 pi/pitch. Here we show that in the limit of a small electric field, E, applied perpendicular to t(0), a solution to the minimizer of the elastic free energy, including a linear coupling between E and splay/bend deformations of n (the flexoelectric term), is one where the director develops a small periodic component parallel to t(0). As the wave number of this distortion is also q, the net effect is a rotation of the optic axis by a small angle relative to t(0). There is no threshold for this effect when the dielectric anisotropy epsilon(a) is greater than epsilon(a) > -8 pi e(2)/K. e is the flexoelectric coefficient and K is an elastic constant. When E parallel to t(0) and epsilon(a) > 0, it is well-known that this director configuration can be created by boundary conditions. In which case, above a critical field, E-c, the cholesteric helix transforms to a uniform director field with n parallel to t(0), without q --> 0 continuously and without introducing defects. As this is similar to solutions presented here when E perpendicular to n but epsilon(a) < 0, the suggestion is that flexoelectricity could mediate a similar commensurate defect free vanishing of the cholesteric helix in this case. When epsilon(a) > 0, the conclusion is that a defect free transformation of the cholesteric helix to a uniform director field with n parallel to E requires the assistance of induced flows.