  We present an approach to solve a Bethe-Salpeter (BS) equation exactly
without any approximation if the kernel of the BS equation exactly is
instantaneous, and take positronium as an example to illustrate the general
features of the solutions. As a middle stage, a set of coupled and
self-consistent integration equations for a few scalar functions can be
equivalently derived from the BS equation always, which are solvable
accurately. For positronium, precise corrections to those of the Schr\"odinger
equation in order $v$ (relative velocity) in eigenfunctions, in order $v^2$ in
eigenvalues, and the possible mixing, such as that between $S$ ($P$) and $D$
($F$) components in $J^{PC}=1^{--}$ ($J^{PC}=2^{++}$) states as well, are
determined quantitatively. Moreover, we also point out that there is a
problematic step in the classical derivation which was proposed first by E.E.
Salpeter. Finally, we emphasize that for the effective theories (such as NRQED
and NRQCD etc) we should pay great attention on the corrections indicated by
the exact solutions.
