We consider the necessary conditions for Nash-points of Vlasov-McKean functionals J(i)[v] = integral(Q)mf(i) (., m, v) dx dt (i = 1, ..., N). The corresponding payoffs f(i) depend on the controls and, in addition, on the field variable m = m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.