Clear[ p, by, size ]
by=0.406 ;
size = 800 ;

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_]:=ImageResize[Rasterize[g,"Image",ImageResolution->72*2,Background->None],Scaled[1/2]]

Clear[img]
img=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,ImageScaled[{1/2, 1/2, 0}],{Center,Center},size],po,ImageSize->size]

(* manual fiddling with the scaling and size *)
Manipulate[
Graphics3D[Inset[img,ImageScaled[{s, t, u}],{Center,Center},sz],po,ImageSize->sz]
,{{s, 0.505}, 0, 1}
,{{t, 0.414}, 0, 1}
,{{u, 0.598}, 0, 1}
,{{sz, 1124}, 100, 1200}
]