  Let X be a smooth real algebraic variety. Let $\xi$ be a distribution on it.
One can define the singular support of $\xi$ to be the singular support of the
$D_X$-module generated by $\xi$ (some times it is also called the
characteristic variety). A powerful property of the singular support is that it
is a coisotropic subvariety of $T^*X$. This is the integrability theorem (see
[KKS, Mal, Gab]). This theorem turned out to be useful in representation theory
of real reductive groups (see e.g. [AG_AMOT, AS, Say]).
  The aim of this paper is to give an analog of this theorem to the
non-Archimedean case. The theory of D-modules is not available to us so we need
a different definition of the singular support. We use the notion wave front
set from [Hef] and define the singular support to be its Zariski closure. Then
we prove that the singular support satisfies some property that we call weakly
coisotropic, which is weaker than being coisotropic but is enough for some
applications. We also prove some other properties of the singular support that
were trivial in the Archimedean case (using the algebraic definition) but not
obvious in the non-Archimedean case.
