G=ρ=M=1/Sqrt[2Pi] (* Bezwymiarowe jednostki, (Ω^2)/(2*π*G*ρ) *)

Sqrt[2*Pi*G*ρ]

(*Clear[Ω,G,c,M,R]*)

(*a=b=c=R*)

r={x,y,z}

o={a,b,c}

(*a b c Integrate[1/(aa[[1]]^2+u)/Sqrt[(a^2+u)(b^2+u)(c^2+u)],{u,0,Infinity},Assumptions->a>0&&b>0&&c>0]*)

Integrate[1/(a^2+u)/Sqrt[(a^2+u)(b^2+u)(c^2+u)],{u,0,Infinity}]

Ax[a_?NumericQ,b_?NumericQ,c_?NumericQ]:= 
a b c NIntegrate[1/(a^2+u)/Sqrt[(a^2+u)(b^2+u)(c^2+u)],{u,0,Infinity},Method->"DoubleExponential"(*,WorkingPrecision->32*)]

Ay[a_?NumericQ,b_?NumericQ,c_?NumericQ]:= 
a b c NIntegrate[1/(b^2+u)/Sqrt[(a^2+u)(b^2+u)(c^2+u)],{u,0,Infinity},Method->"DoubleExponential"(*,WorkingPrecision->32*)]

Az[a_?NumericQ,b_?NumericQ,c_?NumericQ]:= 
a b c NIntegrate[1/(c^2+u)/Sqrt[(a^2+u)(b^2+u)(c^2+u)],{u,0,Infinity},Method->"DoubleExponential"(*,WorkingPrecision->32*)]

A={Ax,Ay,Az}

A[[1]][1,2,3]

Φc=1/2 Ω^2 (x^2+y^2)

Φg=Pi G ρ Sum[(o[[i]]^2-r[[i]]^2) A[[i]][a,b,c],{i,1,3}]

const=(Φg+Φc/.{x->0,y->0,z->c}//Simplify)

ellipsoid=Φg+Φc-const

α=D[ellipsoid,{x,2}]/2

β=D[ellipsoid,{y,2}]/2

γ=D[ellipsoid,{z,2}]/2//Simplify

ζ=-(ellipsoid-α x^2- β y^2- γ z^2//Simplify)

ζ/γ

eq1=a^2==ζ/α//Simplify

eq2=b^2==ζ/β//Simplify

eq3=4/3 Pi a b c ρ == M

R0=(M/(4/3 Pi ρ))^(1/3)

N[R0]

Solve[eq3,c][[1]]

sys={eq1,eq2}/.%//Simplify

Ω=0.4

jacobi={a,b,(3/(4 a b π))}/.FindRoot[Evaluate@sys,{{a,1.25R0},{b,0.75R0}}]

maclaurin={a,b,(3/(4 a b π))}/.FindRoot[Evaluate@sys,{{a,R0},{b,R0}}]

maclaurin/R0

Graphics3D[{Ellipsoid[{0,0,1},maclaurin],Ellipsoid[{0,0,-1},jacobi],{Thick,Line[{{0,0,-2},{0,0,2}}]}}]

Ωtbl=Range[0,1/2,1/100]//Rest//Reverse

N[Ωtbl]

(*Ωtbl2=Range[8/10,1,3/100]*)

R0/(1/2(1-Sqrt[1-4 Ω^2]))^(1/3)/.Ω->0.01

Check

ε1=ParallelTable[
Check[FindRoot[Evaluate@sys,{{a,1.25 R0},{b,0.75R0}},MaxIterations->1024],
FindRoot[Evaluate@sys,{{a, R0},{b,R0}},MaxIterations->1024]]
,{Ω,Ωtbl}];

aTBL={Ωtbl,a/.ε1}//Transpose;
bTBL={Ωtbl,b/.ε1}//Transpose;
cTBL={Ωtbl,M/(4/3Pi ρ a b)/.ε1}//Transpose;

w1=ListPlot[{aTBL,bTBL,cTBL},Joined->True(*,PlotMarkers->Automatic*),PlotRange->{0.5{0,1},2{0,1}},PlotStyle->{Red,{Blue,Dashed},{Black,Dashed}}]