  In this paper, the general perturbation problem of piecewise smooth
integrable differential systems with two switching planes is considered.
Firstly, when the unperturbed system has a family of periodic orbits, the first
order Melnikov vector function is derived which can be used to study the number
of periodic orbits bifurcated from the period annuli. Then, by using the
obtained Melnikov vector function, we get an upper bound of the number of
periodic orbits of a concrete $n$-dimensional piecewise smooth differential
system.
