Clear[ p, by, size ]
by=0.406 ;
size = 500 ;

p = ContourPlot3D[ 
-Cos[u]+by Cos[v]==2 e (Sqrt[(by^2)+1])
, {u, - Pi,  Pi}
, {v, - Pi,  Pi}
, {e, -1,  1}
(*, PlotLegends -> l (*Placed[ l , {Right,Top} ]*)*)
, Ticks -> {{-Pi, -Pi/2, 0, Pi/2, Pi}, {-Pi, -Pi/2, 0, Pi/2, Pi}, None}
,ImageSize->size
]

Clear[antialias]
antialias[g_]:=Rasterize[g,{"Graphics","Image", "RasterSize"},RasterSize -> size*2,Background->None]
(*Rasterize[g,"Image",RasterSize -> size,Background->None] // FullForm*)

Clear[gg, img, rs]
{gg, img, rs}=antialias[Show[p,Axes->False,Boxed->False]] 

Clear[plotopt]
plotopt::usage = "Extract the options for the plot p, retaining symbolic Tick values if set." ;
plotopt[p_] := Module[{t, ta},
t = Ticks /. Options[p] ;
ta = Ticks /. AbsoluteOptions[p] ;
AbsoluteOptions[p] /. ta -> t
]

Clear[po]
po = plotopt[p] ;

Graphics3D[Inset[img],po(*,ImageSize->size*)]

Graphics3D[Inset[gg],po(*,ImageSize->size*)]