  We present and develop a general dispersive framework allowing us to
construct representations of the amplitudes for the processes $P\pi\to\pi\pi$,
$P=K,\eta$, valid at the two-loop level in the low-energy expansion. The
construction proceeds through a two-step iteration, starting from the
tree-level amplitudes and their S and P partial-wave projections. The one-loop
amplitudes are obtained for all possible configurations of pion masses. The
second iteration is presented in detail in the cases where either all masses of
charged and neutral pions are equal, or for the decay into three neutral pions.
Issues related to analyticity properties of the amplitudes and of their lowest
partial-wave projections are given particular attention. This study is
introduced by a brief survey of the situation, for both experimental and
theoretical aspects, of the decay modes into three pions of charged and neutral
kaons and of the eta meson.
