Differential Equation Involving a Bivariate PDF and its Marginal CDF

Question

I have a differential equation of the form $$ P(x_1,0) = R(x_1) Q(x_1), $$ where $P$ is an unknown, isotropic bivariate probability density function (pdf) i.e. $$ P(x_1,x_2) = P(x_3,x_4), \quad \forall x_1,x_2,x_3,x_4 \in \mathbb{R} \; | \; x_1^2 + x_2^2 = x_3^2 + x_4^2, $$ $R$ a continuous function, and $Q$ is the distribution function (cdf) of the marginal pdf of $P$ i.e., $$ Q(x_1) = \int_{-\infty}^{\infty} \int_{-\infty}^{x_1} P(t_1,t_2) dt_1 dt_2. $$ If we write $P(x_1,0) = P(x_1) P(0)$, then we fall back to a simple first order differential equation that has a solution of the form $$ P(x) = e^{-\int R(x) dx}. $$ However, I would like to know if the general case can be solved as well, and how.