Phase portrait of system of nonlinear ODEs How can we sketch by hand the phase portrait of a system of nonlinear ODEs like the following?
$$\begin{align} \dot{x} &= 2 - 8x^2-2y^2\\ \dot{y} &= 6xy\end{align}$$
I can easily find the equilibria, which are
$$\left\{ (0, \pm 1), \left(\pm \frac{1}{2}, 0\right) \right\}$$
The corresponding stable subspace for $\left(\pm \frac{1}{2}, 0\right)$ is
$$\mbox{span} \left\{ \left(\frac{2i}{\sqrt{6}}, 1 \right), \left(-\frac{2i}{\sqrt{6}}, 1 \right) \right\}$$
and the unstable subspace for $(0, \pm 1)$ is
$$\mbox{span} \left\{ (0, 1), (1, 0) \right\}$$
respectively. But I can't see how to use these pieces of information to sketch the phase portrait. Any help would really be appreciated!

The basic process is to find the critical points, evaluate each critical point by finding eigenvalues/eigenvectors using the Jacobian, determine and plot $x$ and $y$ nullclines, plot some direction fields and use all of this type of information to draw the phase portrait.
You can see two different views of this process at this website and notes.
For your particular problem
$$x' = 2 - 8x^2-2y^2 \\ y' = 6xy$$
We find the critical points where we simultaneously get $x' = 0, y' = 0$ so
$$(x, y) = (0, -1), (0, 1), \left(-\dfrac{1}{2}, 0\right), \left(\dfrac{1}{2}, 0\right)$$
The Jacobian is 
$$J(x, y) = \begin{bmatrix}\dfrac{\partial x'}{\partial x} & \dfrac{\partial x'}{\partial y}\\\dfrac{\partial y'}{\partial x} & \dfrac{\partial y'}{\partial y}\end{bmatrix} =  \begin{bmatrix}-16 x & -4y\\6y & 6x\end{bmatrix}$$
Evaluate eigenvalue/eigenvector for each critical point
$J(0, -1) \implies \lambda_{1,2} = \pm 2 i \sqrt{6}, v_{1,2} = \left(\mp i \sqrt{\frac{2}{3}}, 1\right) \implies$ spiral
$J(0, 1) \implies \lambda_{1,2} = \pm 2 i \sqrt{6}, v_{1,2} = \left(\pm i \sqrt{\frac{2}{3}}, 1\right) \implies$ spiral
$J(-\frac{1}{2}, 0) \implies \lambda_{1,2} = (8, -3), v_{1} = (1,0), v_2 = (0, 1) \implies$ saddle
$J(\frac{1}{2}, 0) \implies \lambda_{1,2} = (-8, 3), v_{1} = (1,0), v_2 = (0, 1) \implies$ saddle
Using all the above (critical points, eigenvalues/eigenvectors, x-nullcline (red and black curves), y-nullcline (green curve), direction fields, etc.), you can now sketch the phase portrait. Exercise - make sure to add direction fields from the two sets of notes linked above so you understand how to do that. The phase portrait will look like:

