  This paper deals with the problem of limit cycle bifurcations for piecewise
smooth integrable differential systems with four zones. When the unperturbed
system has a family of periodic orbits, the first order Melnikov function is
derived which can be used to study the number of limit cycles bifurcated from
the periodic orbits. As an application, using the first order Melnikov function
and Picard-Fuchs equation, we obtain an upper bound of the number of bifurcated
limit cycles of a concrete piecewise smooth differential system.
