Possible Close-form solution for 2 dimensional $\frac{d R}{dt}R^T = \mathbf{F}R_t R_t^T + \frac{1}{2} \mathbf{G}\mathbf{G}^T$

Question

I want to solve $\mathbf{R}_t$ for $$ \frac{d \mathbf{R}_t}{dt}\mathbf{R}_t^T = \mathbf{F}\mathbf{R}_t \mathbf{R}_t^T + \frac{1}{2} \mathbf{G}\mathbf{G}^T, $$ where $\mathbf{F},\mathbf{G}$ are 2x2 matrix and assume init condition $\mathbf{R}_0 \mathbf{R}_0^T$ is a known PSD matrix.

A related question is this one, where $\Sigma_t = \mathbf{R}_t \mathbf{R}_t^T$ and it is easy to check that $$ \frac{d \Sigma_t}{dt} = \mathbf{F}\Sigma_t + \Sigma_t \mathbf{F}^T + \mathbf{G}\mathbf{G}^T, $$ where we almost have closed-form for $\Sigma_t$ (see answers in the question).

I have the following two questions

1. Do we have a nice closed-form solution for $\mathbf{R}_t$?
2. If not, how can we solve the equation with good accuracy? We should note that $\Sigma_t = \mathbf{R}_t \mathbf{R}_t^T$ has closed-form and is PSD, if we just simply use Euler integral to get numerical $\mathbf{R}_t$ we can not guarantee $\Sigma_t = \mathbf{R}_t \mathbf{R}_t^T$?