cp1=ContourPlot3D[z==x^4+y^3-1,
{x,-2,2},{y,-2,2},{z,-1,1},
ContourStyle->{Blue,Opacity[0.2]},
Mesh->20,MeshStyle->{Darker[Blue],Blue},
MeshFunctions->{Sin[2#1*#3]&,Cos[3#2*#3]&}];
cp2=Graphics3D[{Opacity[.5],EdgeForm[Darker[Blue]],
Translate[Icosahedron[0.3],
Table[{x,(1-x^4)^(1/3),0},{x,Range[-1,1,0.5]}]]}];
Show[cp1,cp2,PlotRange->All,Axes->False,
Boxed->False,PlotLabel->"Surfaces' Translation"]
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