Parametrization of helix on a 3D quatrefoil

Question

The problem given is to parametrize a special helical coil which sits on a surface of a (nameless?) shape which is a surface of revolution, where the generatrix is the meeting of two Bernoulli lemniscates. Thus the generatrix is a 2D Four-leaf clover or rose (aka quatrefoil curve) while the directrix is a closed curve(circle) and axis of revolution is the Oz axis (see attached pictures). (Could be one lemniscate or two, the shape generated is the same)

Managed to build so far a helix sitting on such a (nameless?) surface in 3ds maxscript as seen in attached pictures, but the parametrization is not how it needs to be.

Pseudo-code:
for t=-90 to 270 do (
typical parametric equations for a Bernoulli lemniscate rotated 45 degrees in the xOz plane:
     cst=(a*cos(t)/(sin(t)^2+1) 
     x=cst*(1+sin(t))        
     y=cst*(1-sin(t))
     Place blue point [x, 0 ,y] 
    )
for t=0 to 360 do (
same as above:
     cst=(a*cos(t)/(sin(t)^2+1)
     x=cst*(1+sin(t))        
     y=cst*(1-sin(t))
now the helix parametric equations:
     helix_x=x*cos(helix_turns*t)
     helix_y=x*sin(helix_turns*t)
     helix_z=y 
     Place yellow point [helix_x, helix_y, helix_z]         
    )

Can provide further details for the equations above but it’s beside the point, since they’re not the required helix. Excuse the inappropriate terminology, but the required helix must have a variable pitch in such a way that acts with a bias(more turns) towards the tips(ends) and middle and less turns on the outside (where obviously the helix path is longer), such that a point charge particle(sphere) attached to the helix path moves linearly (with constant speed) both throughout the rotational cross-section (lemniscate) and through the helix at every instance in time.

Again, attach a particle to the helix. As the particle moves through the helix, so is its cross-sectional plane around the Oz axis, such that the particle moves through the helix path BUT ALSO through the lemniscate path in the cross-section plane. Parametrize helix such that the movement through both the lemniscate path and helix path is uniform, linear and with a constant speed in time. Excuse if I'm confusing terms. I hope I'm being understood.

Being a little rusty in differential geometry, be patient with me. Any help is better than no help, and it will be greatly appreciated. Thanks!