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We explore the existence of a continuous marginal law with respect to theLebesgue measure for each component $(X,Y,Z)$ of the solution to coupledquadratic forward-backward stochastic differential equations (QFBSDEs) {forwhich the drift coefficient of the forward component is either bounded andmeasurable or H\"older continuous. Our approach relies on a combination of theexistence of a weak decoupling field, the integration with respect to spacetime local time (see \cite{Ein2006}), the analysis of the backward Kolmogorovequation associated to the forward component along with an It\^o-Tanaka trick(see \cite{FlanGubiPrio10})}. The framework of this paper is beyond allexisting papers on density analysis for Markovian BSDEs and constitutes a majorrefinement of the exisiting results. As an independent result, we also derive acomonotonicity theorem for the control variable in this frame and thusextending the works \cite{ChenKulWei05}, \cite{DosDos13}.

Density Analysis for coupled forward-backward SDEs with non-Lipschitz drifts