Intersection of saddle surface and cone

a=Plot3D[x y,{x,-3,3},{y,-3,3},Mesh->None,PlotStyle->{Opacity[0.5],Blue},BoxRatios->{1,1,1},PlotRange->2,ClippingStyle->None];

b=Plot3D[{Sqrt[x  y],-Sqrt[x y]},{x,-3,3},{y,-3,3},Mesh->None,PlotStyle->{{Green},{Green}},BoxRatios->{1,1,1},RegionFunction->Function[{x,y},Abs[x+y]≤3]];

Show[Graphics[a],Graphics[b],AspectRatio->Automatic]

Family of cubics

Manipulate[Plot[{Sqrt[x^3-x+t],-Sqrt[x^3-x+t]},{x,-2,2},AspectRatio->Automatic,PlotRange->{{-4,4},{-4,4}},PlotStyle->Thick],{t,-2,2}]

cubic=ParametricPlot3D[{{x,Sqrt[x^3-x+t],t},
{x,-Sqrt[x^3-x+t],t}},{x,-2,2},{t,-2,2},AxesLabel->{"x","y","t"},Mesh->None,PlotStyle->{{Opacity[0.4],Blue},{Opacity[0.4],Blue}}];

Manipulate[Show[cubic,ParametricPlot3D[{x,t,s},{x,-2,2},{t,-2,2},Mesh->None,PlotStyle->Red]],{s,-2,2}]