In this paper the field emission triode system with the axially symmetric electrodes is considered. The field emitter and anode are the confocal paraboloidal surfaces of revolution, modulator is a part of the confocal paraboloidal surface of revolution. To solve the boundary-value problem for the Laplace’s equation with the boundary conditions of the first kind the paraboloidal coordinates are used. Dual integral equations method is applied to calculate the axisymmetric electrostatic field. The electrostatic potential distribution over the entire region of the triode field system is found in the form of expansions in Bessel functions. Unknown expansion coefficients determination is reduced to solving the Fredholm equation of the second kind. All geometrical dimensions of the system are the parameters of the problem.