f = \[Omega]^2 c^2 e1 e2 e3 (\[Omega]^4 c^4 - \[Omega]^2 c^2 ((kx^2 + ky^2)/e3 + (kx^2 + kz^2)/e2 + (ky^2 + kz^2) e1) + (kx^2/(e2 e3) + ky^2/(e1 e3) + kz^2/(e1 e2)) (kx^2 + 
         ky^2 + kz^2)) /. {c -> 1, e1 -> 1, e2 -> 2, e3 -> 3, \[Omega] -> 1};
p1 = Table[
   Show[
    ContourPlot3D[{f == 0}, {kx, -2, 2}, {ky, -2, 2}, {kz, -2, 2}, ContourStyle -> Directive[Opacity[0.5], Orange], AxesLabel -> {"\!\(\*SubscriptBox[\(k\), \(x\)]\)", "\!\(\*SubscriptBox[\(k\), \(y\)]\)", 
       "\!\(\*SubscriptBox[\(k\), \(z\)]\)"}, Mesh -> None, Ticks -> None, LabelStyle -> {Bold, Black}, RegionFunction -> Function[{kx, ky, kz}, kx < a], BoxRatios -> {1, 1, 1}]
    ,
    Graphics3D[{Black, Thick, Line[{2*{-1.24, 0, -0.68}, 2*{1.24, 0, 0.68}}], Line[{2*{-1.24, 0, 0.68}, 2*{1.24, 0, -0.68}}]}]
    ]
   , {a, 1.8, 1.24, -0.05}];
p2 = Show[
   ContourPlot3D[{f == 0}, {kx, -2, 2}, {ky, -2, 2}, {kz, -2, 2}, ContourStyle -> Directive[Opacity[0.5], Orange], AxesLabel -> {"\!\(\*SubscriptBox[\(k\), \(x\)]\)", "\!\(\*SubscriptBox[\(k\), \(y\)]\)", 
      "\!\(\*SubscriptBox[\(k\), \(z\)]\)"}, Mesh -> None, Ticks -> None, LabelStyle -> {Bold, Black}, RegionFunction -> Function[{kx, ky, kz}, kx < 1.24], BoxRatios -> {1, 1, 1}]
   ,
   Graphics3D[{Black, Thick, Line[{2*{-1.24, 0, -0.68}, 2*{1.24, 0, 0.68}}], Line[{2*{-1.24, 0, 0.68}, 2*{1.24, 0, -0.68}}]}]
   ];
p3 = Table[
   Show[
    ContourPlot3D[{f == 0}, {kx, -2, 2}, {ky, -2, 2}, {kz, -2, 2}, ContourStyle -> Directive[Opacity[0.5], Orange], AxesLabel -> {"\!\(\*SubscriptBox[\(k\), \(x\)]\)", "\!\(\*SubscriptBox[\(k\), \(y\)]\)", 
       "\!\(\*SubscriptBox[\(k\), \(z\)]\)"}, Mesh -> None, Ticks -> None, LabelStyle -> {Bold, Black}, RegionFunction -> Function[{kx, ky, kz}, kx < a], BoxRatios -> {1, 1, 1}]
    ,
    Graphics3D[{Black, Thick, Line[{2*{-1.24, 0, -0.68}, 2*{1.24, 0, 0.68}}], Line[{2*{-1.24, 0, 0.68}, 2*{1.24, 0, -0.68}}]}]
    ]
   , {a, 1.24, 0.5, -0.05}];
ListAnimate[
 Join[p1, Table[p2, 5], p3, Reverse[p3], Table[p2, 5], Reverse[p1] ] ]

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