Embedded Spheres with an (0,1,n) and (0,1,2) end

ω1[z_] :=1/z^2
ω2[z_] :=1/z;
ω3[z_] :=I+z^n;

Integrate[{ω1[z], ω2[z], ω3[z]},z]

f[r_,t_]=ComplexExpand[Re[Integrate[{ω1[z], ω2[z], ω3[z]},z]/.z-> r E^(I t)],TargetFunctions->{Re,Im}]//Simplify

f[r_,t_]:={-((Cos[t])/r),Log[r],r ((((r^n) Cos[(1+n) t])/(1+n))-Sin[t])}

ParametricPlot3D[f[r,t]/{6,2,2}/.n->1,{r,.15,1.5},{t,0,2Pi},PlotRange->All, AxesLabel->{x,y,z}]

ParametricPlot3D[f[r,t]/{4,2,2}/.n->4,{r,.15,1.5},{t,0,2Pi},PlotRange->All, AxesLabel->{x,y,z}]

Embeddednes and Regularity

g[x_,y_]=ComplexExpand[Re[Integrate[{ω1[z], ω2[z], ω3[z]},z]/.z-> x+I y],TargetFunctions->{Re,Im}]//Simplify

g[x,y1]-g[x,y2]//Factor

g[x,y]-g[x,-y]//Factor

Cross@@Transpose[D[g[x,y],{{x,y}}]]

%/.y->0