Full line trajectories plot for the solution of Second Order nonlinear coupled differential equations

I wanted to plot a phase plane containing the trajectories of the solutions found by using ‘NDSolve’ using the initial conditions for x[0], y[0], x'[0] and y'[0]. The equations are: x”[t] – 2 y'[t] == -x[t] + y[t]^2; y”[t] + 2 x'[t] == x[t] + y[t] + x[t]*y[t]

The equilibrium point for the system is (0,0). I have plotted the stream plot for the system but unable to plot a phase portrait that would give me the full line trajectories of the system for different initial conditions. I am also looking for any periodic solution if present in it. The stream plot I got is given below and I would take initial conditions from it.

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I get this by using the Parametric Plot of the NDSolve solution:

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Kindly help in this capacity. Thanks in advance.