The Pontencial,

ContourPlot[(1/2)((x^2)+(y^2)+2 (x^2)y-(2/3)(y^3)),{x,-1.2,1.2},{y,-1.2,1.2},Contours->{2.4,2.2,2,1.8,1.6,1.4,1.2,1,1/2,1/4,1/6,1/8,1/12,1/24,1/100,1/1000},Axes->True,Frame->False]
Plot3D[(1/2)((x^2)+(y^2)+2 (x^2)y-(2/3)(y^3)),{x,-1.2,1.2},{y,-1.2,1.2},MeshFunctions->{#3&},Mesh->15,ColorFunction->(ColorData["DarkRainbow"][#3]&)]
Plot3D[(1/2)((x^2)+(y^2)+2 (x^2)y-(2/3)(y^3)),{x,-.4,.4},{y,-.4,.4},MeshFunctions->{#3&},Mesh->15,ColorFunction->(ColorData["DarkRainbow"][#3]&)]
Plot3D[(1/2)((x^2)+(y^2)+2 (x^2)y-(2/3)(y^3)),{x,-25,25},{y,-25,25},MeshFunctions->{#3&},Mesh->15,ColorFunction->(ColorData["DarkRainbow"][#3]&)]

v[y_,ee_]:=(Sqrt[((2 ee-(y^2)+(2/3)(y^3))/(1+2 y))])

Plot[{Table[{v[x,i],-v[x,i]},{i,{1/6,1/8,1/12,1/24,1/100}}]},{x,-1.5,2.5},PlotRange->{{-1.5,2.5},{-2,2}},AxesLabel->{"y","x"}]

The Jacobian

j=(({{0, 0, 1, 0}, {0, 0, 0, 1}, {-1-2y, -2x, 0, 0}, {-2x, -1-2y, 0, 0}}))

Eigenvalues[j]

Eigenvectors[j]

Clear[vv,ff]

The Potential and it's derivative

vv[{x_,y_}]:=(1/2)((x^2)+(y^2)+2 (x^2)y-(2/3)(y^3))

D[vv[{x,y}],{{x,y}}]

a=Solve[D[vv[{x,y}],{{x,y}}]==0,{x,y}]

Extremum points

{x, y} /. a

The value of the potential at those points

Map[vv,{x,y}/.a]

2nd derivative

D[vv[{x,y}],{{x,y},2}][0,0]

{{(1/2) (2+4 y),2 x},{2 x,(1/2) (2-4 y)}}

2nd derivative discriminant

ff[{x_,y_}]:=(1/2) (2+4 y) (1/2) (2-4 y) - 4 (x^2)

Map[ff,{x,y}/.a]

{1,-3,-3,-3}

(1/2) (2+4 0)

2nd derivative discriminant
the points {0,1},{-((Sqrt[3])/2),-(1/2)},{((Sqrt[3])/2),-(1/2)} are saddle points
and {0, 0} is a relative minimum

fe[ee_,y_]:=((2 (y^3))/3.)+2. ee-(y^2)

Plot[fe[1/12,y],{y,-0.7,0.8},AxesOrigin->{0,0}]

Plot[fe[1/6,y],{y,-0.7,1.4},AxesOrigin->{0,0}]

Plot[fe[1/4,y],{y,-0.7,1.4},AxesOrigin->{0,0}]