  In this paper we prove a necessary condition of the optimal control problem
for a class of general mean-field forward-backward stochastic systems with
jumps in the case where the diffusion coefficients depend on control, the
control set does not need to be convex, the coefficients of jump terms are
independent of control as well as the coefficients of mean-field backward
stochastic differential equations depend on the joint law of $(X(t),Y(t))$. Two
new adjoint equations are brought in as well as several new generic estimates
of their solutions are investigated for analysing the higher terms, especially,
those involving the expectation which come from the derivatives of the
coefficients with respect to the measure. Utilizing these subtle estimates, the
second-order expansion of the cost functional, which is the key point to
analyse the necessary condition, is obtained, and whereafter the stochastic
maximum principle.
