a=2;b=3;c=4;

eqns[x0_,z0_]:={a x'[t]==(b-c)y[t]z[t],b y'[t]==(c-a)z[t]x[t],c z'[t]==(a-b)x[t]y[t],x[0]==x0,y[0]==0,z[0]==z0};

T=50;
n=17;
sol=Table[NDSolve[eqns[Cos[Pi p/(2n)]/Sqrt[a],Sin[Pi p/(2n)]/Sqrt[c]],{x[t],y[t],z[t]},{t,0,T}],{p,1,n-1}];

Length[sol]

orbits=ParametricPlot3D[Join[{x[t],y[t],z[t]}/.sol,{-x[t],-y[t],-z[t]}/.sol],{t,0,T},Boxed->False,Axes->False,PlotStyle->{Black,Thickness[.003]}]

surf=ContourPlot3D[a x^2+b y^2+c z^2==.99,{x,-1/Sqrt[a],1/Sqrt[a]},{y,-1/Sqrt[b],1/Sqrt[b]},{z,-1/Sqrt[c],1/Sqrt[c]},BoxRatios->Automatic,Boxed->False,Axes->False,Mesh->False,ContourStyle->Opacity[0.6]]

Show[surf,orbits,ViewPoint->{.8,1.5,.5}]

Export[NotebookDirectory[]<>"wobble.png",%126,ImageSize->1000];
image=Import[NotebookDirectory[]<>"wobble.png"];
Export[NotebookDirectory[]<>"wobble.eps",image];