integrand1[x1_,λ_,ξ_]:=2/Sqrt[λ^2(1-ξ^2)-x1^(-2/λ)*(1-ξ^((2λ-2)/λ))]
endpoints1[λ_,P_]:=Select[x/. NSolve[-2*P==λ^2*x^2-x^(2-2/λ),x,Infinity],NumericQ[#]&&Im[#]==0&&Re[#]>0&][[1]]
Period1[λ_,P_]:=NIntegrate[integrand1[endpoints1[λ,P],λ,ξ],{ξ,0,1}]
integrand2[x2_,λ_,ξ_]:=2/Sqrt[λ^2(1-ξ^2)+x2^(-2/λ)*(1-ξ^((2λ-2)/λ))]
endpoints2[λ_,P_]:=Select[x/. NSolve[-2*P==λ^2*x^2+x^(2-2/λ),x,Infinity],NumericQ[#]&&Im[#]==0&&Re[#]>0&][[1]]
Period2[λ_,P_]:=NIntegrate[integrand2[endpoints2[λ,P],λ,ξ],{ξ,0,1}]
Tsum[λ_,P_]:=Period1[λ,P]+Period2[λ,P]

Pstar=-79399495348/1000000000000;
N[endpoints1[5/4,Pstar],30]
N[endpoints2[5/4,Pstar],30]
N[Period1[5/4,Pstar],30]
N[Period2[5/4,Pstar],30]
Tsum[5/4,Pstar]-Pi

sol1=NDSolve[{((λ-1)/λ)*ψ1[θ]^(1-2/(λ))==(λ)^2*ψ1[θ]+ψ1''[θ]/.λ->5/4,ψ1[0]==,ψ1'[0]==0},ψ1,{θ,-(-0.000001)/2,(-0.000001)/2},WorkingPrecision->30];
F1[θ_?NumericQ]:=(ψ1/.sol1[[1]])[θ]

sol2=NDSolve[{-((λ-1)/λ)*Abs[ψ2[θ]]^(1-2/(λ))==(λ)^2*ψ2[θ]+ψ2''[θ]/.λ->5/4,ψ2[0]==,ψ2'[0]==0},ψ2,{θ,-(-0.000001)/2,(-0.000001)/2},WorkingPrecision->30];
F2[θ_?NumericQ]:=(ψ2/.sol2[[1]])[θ]

a=1.496811767622862276994055719114840030670166015625;
b=0.07398355917097741707433300462071201764047145843505859375;
Final[θ_?NumericQ]:=Piecewise[{{F1[a-θ],0<=θ<=a},{F1[θ-a],a<θ<=2*a},{-F2[2*a+b-θ],2*a<θ<2*a+b},{-F2[θ-(2*a+b)],2*a+b<=θ<=2*a+2*b}}]
FinalP[θ_?NumericQ]:=Final[Mod[θ,2*a+2*b]]
FinalD[θ_?NumericQ]:=5000*(FinalP[θ+1/10000]-FinalP[θ-1/10000])
Plot[FinalP[θ],{θ,0,10},PlotRange->All]
ParametricPlot[{FinalP[θ],FinalD[θ]},{θ,0,2Pi}]

vf[x_?NumericQ,y_?NumericQ]:=Module[{r=Sqrt[x^2+y^2],th=ArcTan[x,y]},{-4/3*x+r^(1/4) (-(x FinalD[th])/r-(5y FinalP[th])/(4 r)),-4/3*y+r^(1/4) (-(y FinalD[th])/r+(5 x FinalP[th])/(4 r))}];
r0=(3/4*Sqrt[-2*Pstar])^(4/3);
th0=;
x0=r0*Cos[th0];
y0=-r0*Sin[th0];
VectorPlot[vf[x,y],{x,-1,1},{y,-1,1},PerformanceGoal->"Quality",StreamPoints->2500,AspectRatio->1,MaxRecursion->7 ]

VectorPlot[vf[x,y],{x,0,0.3},{y,-0.06,0},PerformanceGoal->"Quality",StreamPoints->2500,AspectRatio->1,MaxRecursion->7,Epilog->{Red,PointSize[0.015],Point[{x0,y0}]} ]