\( (*This\ program\ takes\ a\ long\ time\ to\ run\ \((about\ 1.5\ hours\ on\ a\ computer\ with\ a\ Pentium\ III\ chip)\) . \ So\ if\ you\ want\ to\ get\ the\ quick\ result\ \((with\ coarser\ resolution)\), \ 
        change\ "Plotpoints"\ option\ in\ "ParametricPlot3D"\ to\ something\ on\ the\ order\ of\ \(\("PlotPoints->{10,20}"\)\(.\)\)\ \ *) \)

\(z_\&_ := 
        z /. Complex[a_, b_] :> 
            Complex[a, \(-b\)]\)
\[LeftDoubleBracketingBar]z_\[RightDoubleBracketingBar] := 
        Simplify[(Sqrt[z Conjugate[z]])]

Clear[i, α, f, z, x, y, U, A, B, h, f, T, F, G, P, Q, s, 
        f]

p = 
        0;
q = 1;
k = 
        2;
 α = (((k ((E^(q/2)) + (E^(-q/2))) - Sqrt[k^2 ((E^(q/2)) + (E^(-q/2)))^2 - 4]))/2) // N // 
            Chop;
αbar = Conjugate[α];

f[λ_] = (((λ^2) - (α^2))/(1 - (α^2) (λ^2)));
U[λ_, x_, 
          y_] = (((λ^(-1))  (x + I y) - λ (x - I y))/(2 ( (E^( q/2)) +  (E^(-q/2)))));
F[λ_, x_, y_]=(1/2)(({{(E^U)[λ, x, y] + (E^(-U[λ, x, y])), (E^U)[λ, x, y] - (E^(-U[λ, x, y]))}, {(E^U)[λ, x, y] - (E^(-U[λ, x, y])), (E^U)[λ, x, y] + (E^(-U[λ, x, y]))}}));

A[λ_, x_, 
          y_] = F[λ, x, y][[1, 1]];
B[λ_, 
          x_, y_] = F[λ, x, y][[1, 
            2]];
sig2=(({{0, -I}, {I,  0}}));
sig3=(({{1, 0}, {0, -1}}));
(Subscript[π, 1])=(({{1, 0}, {0, 0}}));
(Subscript[π, 2])=(({{0, 0}, {0, 1}}));
\(π\&~[λ_, x_, y_]\)=1/(\[LeftDoubleBracketingBar]A[α, x, 
                      
                      y]\[RightDoubleBracketingBar]^2 + \[LeftDoubleBracketingBar]α(\[RightDoubleBracketingBar]^(-2)) \[LeftDoubleBracketingBar]B[α, x, y](\[RightDoubleBracketingBar]^2)) (({{\[LeftDoubleBracketingBar]A[α, x, 
                          y](\[RightDoubleBracketingBar]^2), \(λ\ \(α\^\(-1\)\) \(\(A[α, x, y]\)\&_\) B[α, x, y]\)}, {\(λ\^\(-1\)\ \(\(α\^\(-1\)\)\&_\) 
                    A[α, x, 
                      y] \(B[α, x, y]\)\&_\), \[LeftDoubleBracketingBar]α(\[RightDoubleBracketingBar]^(-2)) \[LeftDoubleBracketingBar]B[α, x, y](\[RightDoubleBracketingBar]^2)}}));
\(\(π\^*\)[λ_, x_, y_]\)=1/(\[LeftDoubleBracketingBar]A[α, x, 
                      y]\[RightDoubleBracketingBar]^2 + \[LeftDoubleBracketingBar]α(\[RightDoubleBracketingBar]^(-2)) \[LeftDoubleBracketingBar]B[α, x, y](\[RightDoubleBracketingBar]^2)) (({{ \[LeftDoubleBracketingBar]α(\[RightDoubleBracketingBar]^(-2)) \[LeftDoubleBracketingBar]B[α, x, y](\[RightDoubleBracketingBar]^2), \(\(-λ\)\ \(α\^\(-1\)\) \(\(A[α, x, y]\)\&_\) B[α, x, y]\)}, {\(\(-λ\^\(-1\)\)\ \(\(α\^\(-1\)\)\&_\) 
                    A[α, x, 
                      y] \(B[α, x, y]\)\&_\), \[LeftDoubleBracketingBar]A[α, x, y](\[RightDoubleBracketingBar]^2)}}));
h[λ_] = (1/(Sqrt[f]))[λ] (Subscript[π, 1]) + (Sqrt[f])[λ] (Subscript[π, 2]);
\(h\&~[λ_, x_, 
          y_] = \(\@f[λ]\) π\&~[λ, x, 
              y] + \(1\/\@f[λ]\) \(π\^*\)[λ, x, 
              y];\)
\(T[x_, 
            y_] = \((\(\(A[α, x, y]\)\&_\) 
                B[α, x, 
                  y] \((1 + α\^2)\))\)/\((α \[LeftDoubleBracketingBar]A[α, x, 
                        y]\[RightDoubleBracketingBar]^2 - \ \(α\&_\) \[LeftDoubleBracketingBar]B[α, x, 
                        y]\[RightDoubleBracketingBar]^2)\);\)

G[λ_, x_, y_]=(I/(Sqrt[\[LeftDoubleBracketingBar]T[x, y](\[RightDoubleBracketingBar]^(2 )) + 1]))(({{-1, -λ T[x, y]}, {\(\(-λ\^\(-1\)\)\ \(T[x, y]\)\&_\), 1}}));
\(NF[λ_, x_, y_] = 
        h[λ] . F[λ, x, y] . h\&~[λ, x, y] . 
          G[λ, x, 
            y];\)
\(\(NF\^*\)[λ_, x_, y_] = 
        Conjugate[
          Inverse[NF[λ, x, 
              y]]];\)
\(f[λ_, x_, 
          y_] = \ \(NF[λ, x, 
                y] . {{0, \(-I\)\ E\^\(\(-q\)/2\)\ }, {I\ E\^\(q/2\)\ , 0}} . \(NF\^*\)[λ, x, y] . sig2 // N\) // 
          Chop;\)
r[x_, 
          y_] = (f[(E^(-I p - q/2)), x, y][[1, 
                      1]] + f[(E^(-I p - q/2)), x, 
                      y][[2, 2]])/2 // N // 
          Chop;
s[x_, 
          y_] = (f[(E^(-I p - q/2)), x, y][[1, 
                      2]] + f[(E^(-I p - q/2)), x, 
                      y][[2, 1]])/2 // N // 
          Chop;
t[x_, 
          y_] = I (f[(E^(-I p - q/2)), x, 
                        y][[1, 
                        2]] - f[(E^(-I p - q/2)), x, 
                        y][[2, 1]])/2 // N // 
          Chop;
u[x_, 
            y_] = (f[(E^(-I p - q/2)), x, 
                        y][[1, 
                        1]] - f[(E^(-I p - q/2)), x, 
                        y][[2, 2]])/2 // N // Chop;
  
pr[x_, y_] = s[x, y]/(1 + r[x, y]) // N // 
          Chop;
ps[x_, 
          y_] = t[x, y]/(1 + r[x, y]) // N // 
          Chop;
pt[x_, 
          y_] = u[x, y]/(1 + r[x, y]) // N // Chop;

Graphic = 
        ParametricPlot3D[{pr[x, y] // N // Chop, ps[x, y] // N // 
                  Chop, pt[x, y]} // N // Chop, {x, 0, 4}, {y, 0, 2 Pi}, 
          PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}, 
          PlotPoints -> {26, 51}];