solution to Eqs. (13) may be written as integrations upon the surface velocity and the fluid stress, u ( x ) = 1 8 πµ ∫ S ( t ) G ( x , y ) · ( σ ( y ) · ˆ n ( y )) dS$_{y}$ + 1 8 π ∫ S ( t ) u ( y ) · T ( x , y ) · ˆ n ( y ) dS$_{y}$ (17) G$_{ij}$ ( x , y ) = δ$_{ij}$ | x − y | + ( x$_{i}$ − y$_{i}$ )( x$_{j}$ − y$_{j}$ ) | x − y | 3 , (18) T$_{ijk}$ ( x , y ) = − 6 ( x$_{i}$ − y$_{i}$ )( x$_{j}$ − y$_{j}$ )( x$_{k}$ − y$_{k}$ ) | x − y | 5 , (19) where are the singular Stokeslet and Stresslet tensors, respectively (see Ref. [48]). By introducing a complementary flow u ′ which has the same values of the surface force σ · ˆ n as the flow u on the surface S ( t ), Eq. (17) may be written solely in terms of the second, double-layer integral, where q ( x ) is an unknown density of the singular Stresslet tensor. In the limit as the x approaches the body surface S ( t ), inserting the no-slip condition for the surface velocity there, we find the expression The vertical swimming velocity U = U · ˆz is related to the Stresslet density as (recall that A is the vesicle surface area). Equation (21) is a well-posed Fredholm integral equation of the second kind for the unknown density q ( x ), and has a unique solution. This approach is numerically better conditioned than those based on first-kind equations. The Stresslet integral operator in Eq. (21) has a six-dimensional nullspace corresponding to rigid body motion, and in the presence of external body forces or torques this representation must be closed by a range completion technique (see Ref. [49]). However, in the swimming problem where the deformation velocity u d ( x ) is specified and there are no body forces or torques, Eqs. (21-22) are closed and prescribe uniquely the swimming velocity U . The integrand in Eq. (21) is discontinuous at the singularity but finite, so that the integrals are computed to second-order in the surface mesh element size using a standard trapezoidal quadrature (setting the quadrature weight to zero at the singularity). The axisymmetry of the problem is inserted into the definition of the body surface as well as the density q ( x ). The number of gridpoints 21 U = − 4 π A ∫ S ( t ) ˆz · q ( x ) dS (22) U + u d ( x ) = ∫ S ( t ) ( q ( y ) − q ( x )) · T ( x , y ) · ˆ n ( y ) dS$_{y}$. (21) u ( x ) = ∫ S ( t ) q ( y ) · T ( x , y ) · ˆ n ( y ) dS$_{y}$, (20)