This paper introduces a novel technique for addressing singular kernels in the variational
boundary element formulation. The study presents an expansion centered on two explicit singular
points, enabling the isolation of all relevant singular terms and extending the analytical capabilities to curved elements, thereby broadening the applicability of the formulation. The proposed
method comprehensively addresses all types of singularities and is validated through rigorous and
extensive numerical experimentation on examples with varied geometries, loading conditions,
and boundary conditions. The diversity and inclusivity of these examples highlight the reliability
and effectiveness of the proposed approach in solving two-dimensional elasticity problems.