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A solution form representing the velocity field and pressure for asymmetric three-dimensional (3D) Stokes flows has been constructed in terms of three $k$-harmonically analytic functions. It has also been shown that it uniquely determines an external velocity field vanishing at infinity. With the obtained solution form, problems of 3D Stokes flows due to asymmetric motions of solid bodies of revolution have been reduced to boundary-value problems for the three $k$-harmonically analytic functions, and the resisting force and torque, exerted on bodies in corresponding motions, have been expressed in terms of the $k$-harmonically analytic functions entering the solution form. For regions, in which Laplace's equation admits separation of variables, the boundary-value problems can be solved in closed form via series or integral representations of $k$-harmonically analytic functions in corresponding curvilinear coordinates. This approach has been demonstrated for asymmetric translation and rotation of solid sphere and solid prolate and oblate spheroids. As the second approach, the boundary-value problems have been reduced to integral equations based on Cauchy's integral formula for $k$-harmonically analytic functions. As an illustration, the integral equations have been solved for asymmetric translation and rotation of solid bispheroids and a solid torus of elliptical cross-section for various values of a geometrical parameter.