Kepler problem in parabolic coordinates

[Kate_12]

Homework Statement

kepler problem
H=1/2m(px^2+py^2+pz^2)-k/(x^2+y^2+z^2)^1/2
with parabolic coordinates (a,b,c)
x=sqrt(ab)cos c
y=sqrt(ab)sin c
z=(a-b)/2
1) rewrite H as a function of new canonical variables (a,b,c, pa,pb,pc)
2) Hamilton-Jacobi equation in this coordinate system turns out to be completely separable. Using the Ansatz S=Wa(a)+Wb(b)+Wc(c)-Et, write the partial differential equation for each Wa, Wb, Wc with suitable separation constants.

Relevant Equations

Hamilton Jacobi equation

I solve (1).
But to solve (2), What should be the suitable separation constants?
I am so confused...

E=2/(m*(a+b)) * (a*(dWa/da)^2+b*(dWb/db)^2-k)+l^2/(2mab)
where l(constant) is pc since c is cyclic.

What should I do to solve the problem?

 

[anuttarasammyak]

if you do not mind could you write down H of your solution 1 ?