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Electromagnetic waves carrying orbital angular momentum (OAM and vortex waves) attract much attention due to their specific characteristics and prospects for use in wireless communication, biomedical engineering, and imaging. Vortex waves are complex spatial entities; therefore, their stable generation is a quite complicated task, especially in the radio frequency and terahertz wave domains, considering that the corresponding antennas must be precisely adjusted. Such adjusting is very difficult to achieve for flexible antennas, which are now being actively introduced into practice. Here, we propose a design of a flexible antenna that is able to stably generate waves carrying OAM even when being subjected to some bending. The antenna is composed of a ring-shaped resonator made of a highly conductive graphene film deposited on a thin polydimethylsiloxane substrate. The antenna is operated on two mutually orthogonal degenerate TMm1modes excited by two supply strip lines to generate vortex waves with a desired topological charge. We verify this ability numerically and in a microwave experiment on two antenna prototypes deriving the OAM mode purity by changing the bending radius of the antenna bearing surface, giving it either positive or negative curvature. The obtained experimental results confirm that the antenna has good performance and can be used for flexible electronic devices and communication systems.