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The framework of generalized analytic functions arising from the related potentials (so-called $k$-harmonically analytic functions) has been developed in application to three-dimensional (3D) axially symmetric Stokes flow problems. Cauchy's integral formula for the class of $k$-harmonically analytic functions has been obtained, and series representations for $k$-harmonically analytic functions for the regions exterior to sphere and prolate and oblate spheroids have been derived. As the central result in the developed framework, a solution form representing the velocity field and pressure for 3D axially symmetric Stokes flows has been constructed in terms of two 0-harmonically analytic functions. It has also been shown that it uniquely determines an external velocity field vanishing at infinity. With the obtained solution form, the problem of 3D Stokes flows due to the axially symmetric translation of a solid body of revolution has been reduced to a boundary-value problem for two 0-harmonically analytic functions, and the resisting force exerted on the body has been expressed in terms of a 0-harmonically analytic function entering the solution form. For regions in which Laplace's equation admits separation of variables, the boundary-value problem can be solved analytically via representations of 0-harmonically analytic functions in corresponding curvilinear coordinates. This approach has been demonstrated for the axially symmetric translation of solid sphere and solid prolate and oblate spheroids. As the second approach, the boundary-value problem has been reduced to an integral equation based on Cauchy's integral formula for $k$-harmonically analytic functions. As an illustration, the integral equation has been solved for the axially symmetric translation of solid bispheroids and the solid torus of elliptical cross-section for various values of a geometrical parameter.