A self-intersecting nullcline
Here is an example that shows that our routine can handle a self-intersecting x-nullcline. There is no equilibrium point at the self-intersection (at (0, 0)).
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<<Local`NullclinePlot` NullclinePlot[{x^3 - 4 x y + 2 y^2 + y^3, -1 + x^2 + y}, {x, -2, 2}, {y, -2, 2}, Nullclines -> True, NullclineShading -> True, ShowEquilibria -> True, EquilibriumPointColor -> White];
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A self-intersecting nullcline