(* This draws an orbit under a quasi-joined ends. First we set some constants*) 
C1:=1; v1:=1; k:=1;

lambda[t_] := Exp[C1*t];

(*This is the a generator hyperbolic but acts as a translation on a hemisphere *)

g1[t_]:={{lambda[t]^3, 0, 0, 0}, {0, 1/lambda[t], 0, 0}, {0, 0, 1/lambda[t], 0}, {0, k*t, 0, 1/lambda[t]}}

(*This is the parabolic element *)

n[s_]:= {{1, 1, 0, 1}, {0, 1, 0, 0}, {0, v1 s, 1, 0}, {0, (v1^2 s^2)/2, v1 s, 1}}

(*The point for the orbit *)

U1:={0.3, 0.3, 0.3,1};

(*This moves one fixed point of g1 inside an affine patch *)

TM:=Transpose[{{1,0,0,1}, {0, 0, 1, 0}, {0, 1,0, 0}, {0,0,0,1}}]

(*We find affine coordinates of the orbits *)

f1[t_, s_]:= (TM.g1[t].n[s].U1)[[1]]/(TM.g1[t].n[s].U1)[[4]]; f2[t_, s_]:= (TM.g1[t].n[s].U1)[[2]]/(TM.g1[t].n[s].U1)[[4]];f3[t_, s_]:= (TM.g1[t].n[s].U1)[[3]]/(TM.g1[t].n[s].U1)[[4]];

(*some computations *)

f1[t,s]

f2[t, s]

f3[0.1, 0.1]

(* The first orbit picture... *)

ParametricPlot3D[{f1[t,s],f2[t,s] ,f3[t,s]} , {t, -10, 10},{s, -10, 10}, BoxRatios->{1,1,1}, Mesh->10, AxesLabel->{x, y, z}, PlotRange->Full]

(* The second orbit picture *)

ParametricPlot3D[{f1[t,s],f2[t,s] ,f3[t,s]} , {t, -20, 20},{s, -20, 20}, BoxRatios->{1,1,1}, Mesh->100, AxesLabel->{x,y,z}, PlotRange->Full,LabelStyle->{FontFamily->"Arial",FontSize->18,Black}]

(* Third one ...*)

ParametricPlot3D[{f1[t,s],f2[t,s] ,f3[t,s]} , {t, -30, 30},{s, -20, 20}, BoxRatios->{1,1,1}, Mesh->300, AxesLabel->{x, y, z}, PlotRange->Full]