  We explore the existence of a continuous marginal law with respect to the
Lebesgue measure for each component $(X,Y,Z)$ of the solution to coupled
quadratic forward-backward stochastic differential equations (QFBSDEs) {for
which the drift coefficient of the forward component is either bounded and
measurable or H\"older continuous. Our approach relies on a combination of the
existence of a weak {\it decoupling field} (see \cite{Delarue2}), the
integration with respect to space time local time (see \cite{Ein2006}), the
analysis of the backward Kolmogorov equation associated to the forward
component along with an It\^o-Tanaka trick (see \cite{FlanGubiPrio10})}. The
framework of this paper is beyond all existing papers on density analysis for
Markovian BSDEs and constitutes a major refinement of the existing results. We
also derive a comonotonicity theorem for the control variable in this frame and
thus extending the works \cite{ChenKulWei05}, \cite{DosDos13}. As applications
of our results, we first analyse the regularity of densities of solution to
coupled FBSDEs. In the second example, we consider a regime switching term
structure interest rate models (see for e.g., \cite{ChenMaYin17}) for which the
corresponding FBSDE has discontinuous drift. Our results enables us to: firstly
study classical and Malliavin differentiability of the solutions for such
models, secondly the existence of density of such solutions. Lastly we consider
a pricing and hedging problem of contingent claims on non-tradable underlying,
when the dynamic of the latter is given by a regime switching SDE (i.e., the
drift coefficient is allowed to be discontinuous). We obtain a representation
of the derivative hedge as the weak derivative of the indifference price
function, thus extending the result in \cite{ArImDR10}.
