This paper deals with symmetric variational regularized BIE formulations for the mixed elastostatic boundary-value problem. A direct version (i.e. in terms of unknown boundary displacement and tractions) is first established. The formulation expresses the stationarity of an augmented potential energy functional, thus being truly a variational BIE formulation. Then an indirect version (in terms of unknown fictitious densities) is established from the direct one. Both are expressed using at most weakly singular integrals followed by regular integrals, by means of a combined use of indirect regularization and Stokes theorem. The displacements and tractions are required to be C0,α continuous and piecewise continuous respectively, thus conventional BEM interpolation of any degree may be used. These formulations provide a basis for the numerical solution of 3D elastic problems. The numerical evaluation of the singular integrals that arise in the process is discussed. The formulations presented here provide some understanding of the underlying principles as well as a sound working base for Galerkin boundary element analysis of elastic problems.