How to plot perturbations of a implicit equation obtained from Eliminate

Question

Suppose we have parametrized curve $\gamma(t)$. For instance, consider $\gamma(t)$ as the folium

folium[t_] := {3 t/(1 + t^3), 3 t^2/(1 + t^3)}

Then we obtained the Cartesian equation $f(x,y)=g(x,y)$ of $\gamma$ using Eliminate.

In our example

foliumimplicit := Eliminate[{x, y}==folium[t],t] 

gives us $x^3-3xy=-y^3$.

I would like to plot perturbations $f(x,y)=g(x,y)+\varepsilon$ of the implicit equation obtained. $x^3-3xy=-y^3+\varepsilon$ of the foliumimplicit equation obtained.

For our example, when $\varepsilon=0.1,-0.1$ the graphs of the $x^3-3xy=-y^3+\varepsilon$ are

Answer 1

One way to plot your curve is as a collection of contour plots with different values of e:

ContourPlot[Evaluate[Table[x^3 - 3 x y == e - y^3, {e, -0.1, 0.1, 0.05}]], 
            {x, -2, 2}, {y, -2, 2}]