Following the historical development of the theory of exact solutions for singularity-driven Hele-Shaw flows, this note demonstrates that the problem of two-dimensional viscous sintering preserves quadrature identities. This provides a unified theoretical perspective in which to understand the two separate free boundary problems. The result is established directly from the equations of motion without appeal to conformal mapping theory, although the result underlies the existence of exact conformal mapping solutions. The formulation leads to a concise, closed-form representation of the evolution equations for the parameters in the conformal mapping function. Some examples are given.