Generating zero-isoclines for a coupled ODE

If f and g are independent of t (which they must be, otherwise I don’t think the concept of zero-isocline makes sense?) then you are just asking to draw contours or level sets of the functions f and g:

using Plots
f(x, y) = x^2 + 2y^2
g(x, y) = x - 2y^3

myrange = -2:0.01:2

contour(myrange, myrange, f, levels=[0])
contour!(myrange, myrange, g, levels=[0])

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