  We present a systematic analysis of next-to-next-to-leading-order diagrams
that contribute to the $pp \to pp\pi^0$ production at threshold. Analytic
expressions are given for the effective transition operators, and the relative
importance of various types of diagrams is discussed. The
vertex-correction-type graphs are found to give only small corrections to lower
order graphs in conformity with expectations. By contrast, we find very large
contributions from the two-pion graphs which can be interpreted as a part of
effective $\sigma$-meson exchange diagrams. The recoil correction to the pion
rescattering diagram also turns out to be large.
