(*Example 1*)
g[x_,y_,c_] = -x^3 + 3*y - y^3 -c;
vplot = VectorPlot[{1,x^2/(1 - y^2)},{x,-3,3},{y,-3,3},VectorScale->{0.04,0.04,None},Axes->True];
crvplot = ContourPlot[Evaluate[Table[g[x,y,c]==0,{c,-16,16,4}]],{x,-3,3},{y,-3,3},PlotPoints->40];
Show[vplot,crvplot]

(*Example 2*)
f[x_,y_,c_] = y^2 - 2*y - x^3 - 2*x^2 -2*x-c;
vplot = VectorPlot[{1,(3*x^2+4*x+2)/(2*(y-1))},{x,-3,3},{y,-3,3},VectorScale->{0.04,0.04,None},Axes->True];
crvplot = ContourPlot[Evaluate[Table[f[x,y,c]==0,{c,-13,8,2}]],{x,-3,3},{y,-3,3},PlotPoints->40];
pts = Graphics[{PointSize[0.02],Point[{0,-1}],Point[{-2,1}]},PlotRange->{{-3,3},{-3,3}}];
Show[vplot,crvplot,pts]

f[x_,y_,c_] = y^4+16*y-8*x^2+x^4+c;
vplot = VectorPlot[{1,(4*x-x^3)/(4+y^3)},{x,-4,4},{y,-4,4},VectorScale->{0.04,0.04,None},Axes->True,PlotRange->{{-4,4},{-4,2}}];
crvplot = ContourPlot[Evaluate[Table[f[x,y,c]==0,{c,-27,30,10}]],{x,-5,5},{y,-5,5},PlotPoints->40];
pts = Graphics[{PointSize[0.02],Point[{0,1}],Point[{3.3488,-1.5874}],Point[{-3.3488,-1.5874}]},PlotRange->{{-3,3},{-3,3}}];
Show[vplot,crvplot,pts]