Brilliant implicitPlot routine from https://mathematica.stackexchange.com/users/1871/xzczd. This is same as above but allows multiple values on one contour plot; each contour represents a value of the conserved Hamiltonian. I have added nonlinear part NL to replicate Symon’s plot of figure 6.3 in his “Applied Hamiltonian Dynamics” article AIP Conf. Proc. 249, 277 (1992). See Symon eqn. 6.38.

ClearAll@implicitPlot
implicitPlot[eq_,range__,coordSys_,opt:OptionsPattern[ContourPlot]]:=Module[{coord},With[{plot=ContourPlot[eq,range,opt,PlotRange->All,ContourStyle->{AbsoluteThickness[2]}],trans=#[coord,#2]&[Function,CoordinateTransform[coordSys->"Cartesian",{coord[1],coord[2]}]]/.coord[i_]:>Part[coord,i]},plot/.GraphicsComplex[coord_,rest_]:>GraphicsComplex[trans[coord\[Transpose]]\[Transpose],rest]]]

J[r_,ν_]=(r^2)ν/2;Manipulate[implicitPlot[(ν-(m/n)) J[r,ν]-((a0/n)) (((((J[r,ν])/(2 ν))))^(n/2)) Cos[n γ]+NL ((J[r,ν])^2),{r,0,3},{γ,0,2 π},"Polar",PlotPoints->npl,Contours->Ham  Table[(i/nc),{i,-nc/2,nc/2}]+.000599,PlotRange->{{-2,2},{-2,2}},ColorFunction->"MintColors",PlotLegends->BarLegend[Automatic]],{{n,4,"order"},1,6,1},{{m,1,"m"},1,4,1},{{ν,.24,"(Subscript[ν, x])"},0,2,.01},
{{a0,.0144,"strength"},.0,.05,.001},{{NL,.0,"nonlin"},-.05,.05,.001},{{Ham,0.024,"Contour Range"},0.00,1,.01},{{nc,40,"# Contours"},2,100,1},{{npl,40,"plot points"},10,200,10},AppearanceElements->All]