How to solve a matrix PDE and stop solving when solution becomes singular?


My question consists of two parts:

  1. How do I get mathematica to solve a PDE Matrix system and plot the result? See below for the PDE matrix system. (By plot the result I mean plot the region where the solution $ \Theta$ is nonsingular.)
  2. How do I stop the integration when the solution matrix becomes singular? I know that away from $ (x_1,x_2)=(0,0)$ the solution matrix $ \Theta$ will become singular how do I stop Mathematica from trying to solve pass this point?

The PDE matrix system I am trying to solve is \begin{align} \dot{\Theta}(x_1,x_2)+A&=\lambda \Theta(x_1,x_2) &\quad \text{Equation}\ \Theta(0,0)&=\begin{pmatrix} 1 & -\frac{1}{2} -\frac{\sqrt{3}}{2} \ 1 & -\frac{1}{2}-\frac{1}{2\sqrt{3}}+\frac{2}{\sqrt{3}} \end{pmatrix} &\quad \text{Initial condition} \end{align} I have specified the values of $ \dot{\Theta}(x_1,x_2),A,\lambda$ in the block below:

(* Definitions *) A = {{-(x1^2 - 1), -2 x2 x1 - 1}, {1, 0}} lambda = {{1/2, -Sqrt[3]/2}, {Sqrt[3]/2, 1/2}} (*Value of theta for x1=x2=0 *) ThetaInit = {{1, -1/2 - Sqrt[3]/2}, {1, -1/2 - 1/(2 Sqrt[3]) +      2/Sqrt[3]}} (*Derative of Theta in terms of t. Note \ \frac{dtheta}{dt}=x1'(t)\frac{\partial theta}{\partial x1}+x2'(t) \ \frac{\partial theta}{\partial x2} *) ThetaDot = ( -(x1^2 - 1) x2 - x1) D[Theta[x1, x2], x1] +    x2 D[Theta[x1, x2], x2] 

Notes

  • If you need any clarification please feel free to ask.