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The formation and utilization of a helical electron beam are studied theoretically for a gyrotron with a very low operating voltage in a range 1.5-1.8 kV. Such a gyrotron ('gyrotrino') was earlier proposed for operation inside a magnetic system of an NMR spectrometer with a dynamic nuclear polarization upgrade. Despite the very low voltage, the optimization of the electrode shape can provide velocity and positional electron spreads not exceeding these values for conventional high-voltage gyrotrons. A very small cathode-anode separation makes the gyrotrino very sensitive to thermal expansion of the gun elements that should be compensated by movement of the cathode. Estimations for long-pulse and CW regimes of the gyrotrino operation show that the ion background significantly decreases the reduction of the beam potential and leads to an acceptable drift of the electron cyclotron frequency at the voltage front. A satisfactory thermal load on the waste-beam collector located in a strong uniform magnetic field can be achieved due to the omnidirectional heat flow regime occurring in the case of thin beam footprint. [ABSTRACT FROM AUTHOR]