(This came up while I was studying 4D rotational dynamics. But that's a different topic.)

I have a 3D subset of 6D space, given as the intersection of two spherical cylinders (I'm not sure if this is the right terminology) and an ellipsoid:

x₁² + x₂² + x₃² = c₁²,

x₄² + x₅² + x₆² = c₂²,

(x₁ - x₄)²/(a₁² + a₄²) + (x₂ - x₅)²/(a₂² + a₄²) + (x₃ - x₆)²/(a₃² + a₄²)

+ (x₁ + x₄)²/(a₂² + a₃²) + (x₂ + x₅)²/(a₃² + a₁²) + (x₃ + x₆)²/(a₁² + a₂²) = 1.

What I want to know is whether this 3D subset splits into several pieces. Is it always connected? or is it always disconnected? or does it depend on the shape parameters c₁,c₂,a₁,a₂,a₃,a₄ ?

(I think we shouldn't count degenerate cases, where slightly changing the parameters changes the connectivity.)

For comparison, consider a 1D subset of 3D space given as the intersection of a circular cylinder and an ellipsoid:

x₁² + x₂² = c₁²,

x₁²/a₁² + x₂²/a₂² + x₃²/a₃² = 1.

(WolframAlpha plot.) This always splits into 2 separate curves, except in degenerate cases where the curves meet at 4-way intersections, or where the cylinder is tangent to the ellipsoid everywhere on its equator.