  A two-particle self-consistency is rarely part of mean-field theories. It is,
however, essential for avoiding spurious critical transitions and unphysical
behavior. We present a general scheme for constructing analytically
controllable approximations with self-consistent equations for the two-particle
vertices based on the parquet equations. We explain in detail how to reduce the
full set of parquet equations not to miss quantum criticality in strong
coupling. We further introduce a decoupling of convolutions of the dynamical
variables in the Bethe-Salpeter equations to make them analytically solvable.
We connect the self-energy with the two-particle vertices to satisfy the Ward
identity and the Schwinger-Dyson equation. We discuss the role of the
one-particle self-consistency in making the approximations reliable in the
whole spectrum of the input parameters. We demonstrate the general construction
on the simplest static approximation that we apply to the Kondo behavior of the
single-impurity Anderson model.
