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ASSUMPTIONS AND GENERAL COMMENTS
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For linear quadratic problems, we know that explicit formulas can be obtained. We shall see that we can solve explicitly the Master equation in both cases, and recover all results obtained in the LQ case. Symmetric and nonsymmetric Riccati equations naturally come in the present framework. It is important to keep in mind that in the Master equation, the argument $m$ is a general element of $L^{2}(\mathbb{R}^{n})$ and not necessarily a probability. When we use the value arising from the FP equation, we get of course a probability density, provided that the initial condition is a probability density. To synthesize formulas, we shall use the following notation. If $m\in L^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ then we set

$$m_{1}=\int_{\mathbb{R}^{n}}m(\xi)d\xi,\: y=\int_{\mathbb{R}^{n}}\xi m(\xi)d\xi.\label{eq:5.1}$$

We then take $$\label{eq:5.2}
\begin{split}
f(x,m,v)&=\frac{1}{2}[x^{*}Qx+v^{*}Rv+(x-Sy)^{*}\bar{Q}(x-Sy)];\\
g(x,m,v)&=Ax+\bar{A}y+Bv;\\
h(x,m)&=\frac{1}{2}[x^{*}Q_{T}x+(x-S_{T}y)^{*}\bar{Q}_{T}(x-S_{T}y)];\\
\sigma(x)&=\sigma;\\
\end{split}$$ and hence $a(x)=a=\sigma\sigma^{*}$. We deduce easily $$\label{eq:5.6}
H(x,m,q) =\frac{1}{2}x^{*}(Q+\bar{Q})\, x-x^{*}\bar{Q}Sy+\frac{1}{2}y^{*}S^{*}\bar{Q}Sy- \frac{1}{2}q^{*}BR^{-1}B^{*}q+q^{*}(Ax+\bar{A}y)$$ and $$G(x,m,q)=Ax+\bar{A}y-BR^{-1}B^{*}q.\label{eq:5.7}$$

MEAN FIELD TYPE CONTROL MASTER EQUATION
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We begin with Bellman equation (\[eq:2.7-1\]) $$\label{eq:5.8}
\left\{
\begin{array}{l}
    \dfrac{\partial V}{\partial t}-\displaystyle\int_{\mathbb{R}^{n}}(\mathcal{A}U-\dfrac{1}{2}\beta^{2}\Delta U)(\xi,m,t)m(\xi)d\xi\\
    \qquad+\dfrac{1}{2}\beta^{2}\displaystyle\int_{\mathbb{R}^{n}}\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial U(\xi,m,t)}{\partial m}(\eta)Dm(\xi)Dm(\eta)d\xi d\eta+\displaystyle\int_{\mathbb{R}^{n}}H(\xi,m,DU)m(\xi)d\xi=0.\\
    V(m,T)=\displaystyle\int_{\mathbb{R}^{n}}h(\xi,m)m(\xi)d\xi;
\end{array}
\right.$$ in which $$U(\xi,m,t)=\frac{\partial V(m,t)}{\partial m}(\xi).\label{eq:5.10}$$

We can rewrite (\[eq:5.8\]) under the present linear quadratic setting $$\label{eq:5.11}
\left\{
\begin{array}{l}
    \displaystyle\frac{\partial V}{\partial t}+\int_{\mathbb{R}^{n}}(\frac{1}{2}\text{tr}\, aD_{\xi}^{2}U(\xi,m,t)+\frac{1}{2}\beta^{2}\Delta_{\xi}U(\xi,m,t))m(\xi)d\xi\\
    \qquad+\displaystyle\frac{1}{2}\beta^{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\sum_{i=1}^{n}\frac{\partial}{\partial\xi_{i}}\frac{\partial}{\partial\eta_{i}}\Big(\frac{\partial U(\xi,m,t)}{\partial m}(\eta)\Big)m(\xi)m(\eta)d\xi d\eta\\
    \qquad+\displaystyle\int_{\mathbb{R}^{n}}\bigg[\frac{1}{2}\xi^{*}(Q+\bar{Q})\xi-\xi^{*}\bar{Q}Sy+\frac{1}{2}y^{*}S^{*}\bar{Q}Sy\\
    