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ODEs and phase portraits

I am trying to write a function such that I have the following ODE:
x''(t)+epsilon*(x(t)^2-1)*x'(t)+x(t)==0  [1] where epsilon is to be one
of the parameters in the function. I want to beable to draw out the
phase portraits for the equation for different values of epsilon.

To deduce the phase portraits in a mathematical procedure I multiply
equation[1] by dx(t)/dt and integrate w.r.t. t.  To do this in
Mathematica is trivial, so I'll skip past this - it is the next step
that I would like assistance with:

Q:Is there a way that I can decompose the result of the above (I'll call
it [2]) into the corresponding pair (below) of ODEs to deduce the
trajectory of the phase portrait?

x'(t) = y(t)
y'(t) = f(x,y)
some f(x,y) function of x and y.

Perhaps there is a built in function that will allow me to do this? If
not do, can anybody suggest another way to do this?

Also after plotting the trajectory is there any way to determine the
direction of it in Mathematica?

thanks for you time.

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