R=2;
r=1.2;
{Slider[Dynamic[x0],{-20,20}],Dynamic[x0]}
{Slider[Dynamic[y0],{-20,20}],Dynamic[y0]}
{Slider[Dynamic[z0],{0,40}],Dynamic[z0]}
w=1;
h:=(x^2+y^2+z^2+(R^2-r^2))^2-4*R^2(x^2+y^2)
g:=x0(x^3+x*(y^2+z^2+w^2(R^2-r^2)-2R^2w^2))+y0(y^3+y*(x^2+z^2+w^2(R^2-r^2)-2R^2w^2))+z0(z^3+z(x^2+y^2+w^2(R^2-r^2)))-2R^2w(x^2+y^2)
p1=ContourPlot3D[{h==0,g==0},{x,-3.3,3.3},{y,-3.3,3.3},{z,-3.3,3.3},MeshFunctions->{Function[{x,y,z,f},h-g]},MeshStyle->{{Thick,Blue}},Mesh->{{0}},PerformanceGoal->"Quality",ContourStyle->Directive[Orange,Opacity[0.0],Specularity[White,30]], PlotPoints->50,RotationAction->"Clip",Boxed->False];
p2=ContourPlot3D[h==0,{x,-3.3,3.3},{y,-3.3,3.3},{z,-3.3,3.3},Mesh->None,ContourStyle->Directive[Orange,Opacity[0.6],Specularity[White,30]],RotationAction->"Clip",Boxed->False];
Show[p1,p2]

ContourPlot3D[g==0,{x,-4,4},{y,-4,4},{z,-4,4}]