  Incorporating the zonal spherical function (zsf) problems on real and
$p$-adic hyperbolic planes into a Zakharov-Shabat integrable system setting, we
find a wide class of integrable evolutions which respect the number-theoretic
properties of the zsf problem. This means that at {\it all} times these real
and $p$-adic systems can be unified into an adelic system with an $S$-matrix
which involves (Dirichlet, Langlands, Shimura...) L-functions.
