P1 = Plot3D[ 2 x y/(x^2 + y^2), {x , -1, 1}, {y, -1, 1}, BoxRatios-> Automatic]

P1a = ParametricPlot3D[ {r Cos[t], r Sin[t], 2 Tan[t]/(1 + Tan[t]^2)}, {r, 0, 1}, {t, 0, 2 Pi}]

f2[x_, y_] := x^2 y / (x^4 + y^2)
P2 = Plot3D[ f2[x,y], {x, -1, 1}, {y, -1, 1}]

p[u_, v_] := Module[{c}, 
If[u == 0 && v ≤ 0, Return[{0,0, ((1+v)/(2+2 v+(v^2)))}]];
If[ v ≥  u^2, Return[{u, v, f2[u, v]}], 
c = (v+ 1)u^2/(1 + u^2);
Return[{u, c, f2[u, c]}]]
]

P[u_, v_] := Module[{a,b,c}, 
{a,b,c} = p[u,v];
Return[{a, -b, -c}];
]

G1 = ParametricPlot3D[p[u, v], {u, -1, 1}, {v, -1, 1}];
G2 = ParametricPlot3D[ P[u, v], {u, -1, 1}, {v, -1, 1}];
A1 = Show[G1, G2,  PlotRange -> All ]