Show[ParametricPlot[{((3 (s^2))/(Exp[s](1-s))),-((6 s+((3 (s^2))/(1-s)))/2)},{s,-5,.999},PlotStyle->{Thickness[0.005],Green}],ParametricPlot[{((3 (s^2))/(Exp[s](1-s))),-((6 s+((3 (s^2))/(1-s)))/2)},{s,1.00001,5},PlotStyle->{Thickness[0.005],Green}],Table[ContourPlot[Exp[s] x+2 s y+3 (s^2)==0,{x,-5,5},{y,-5,5},ContourStyle->{GrayLevel[0.7],Thickness[0.001]}],{s,-5,5,.1}],PlotRange->{{-5,5},{-5,5}}]

Show[Table[ContourPlot[ x+(3 (s^2)-1) y-s-((s^3)-s)(3 (s^2)-1)==0,{x,-5,5},{y,-5,5},ContourStyle->{GrayLevel[0.7],Thickness[0.001]}],{s,-5,5,.01}],Plot[x^3-x,{x,-5,5},PlotStyle->{Thickness[0.005],Blue}],ParametricPlot[{s-((3 (s^2)-1)((1+((((3 (s^2)-1))^2)))/(6 s))),((s^3)-s)+((1+((((3 (s^2)-1))^2)))/(6 s))},{s,-5,-0.00001},PlotStyle->{Thickness[0.009],Green}],ParametricPlot[{s-((3 (s^2)-1)((1+((((3 (s^2)-1))^2)))/(6 s))),((s^3)-s)+((1+((((3 (s^2)-1))^2)))/(6 s))},{s,0.00001,5},PlotStyle->{Thickness[0.009],Green}],PlotRange->{{-5,5},{-5,5}}]