  In this paper, we give several new results on solvability of a quadratic BSDE
whose generator depends also on the mean of both variables. First, we consider
such a BSDE using John-Nirenberg's inequality for BMO martingales to estimate
its contribution to the evolution of the first unknown variable. Then we
consider the BSDE having an additive expected value of a quadratic generator in
addition to the usual quadratic one. In this case, we use a deterministic shift
transformation to the first unknown variable, when the usual quadratic
generator depends neither on the first variable nor its mean, the general case
can be treated by a fixed point argument.
