Zeta is useful occasionally when Opener has a very long solid suit of their own, no fit for Responder's suit, and they don't really care whether or not Responder can support their suit or not. Zeta sets the trump suit unconditionally and at the same time it asks for general Controls in exactly the same way as Beta.
In practice there are very few hands on which it is right to use Zeta. A completely self-sufficient trump suit (eg: AKQJxxx or KQJ10xxxx) is only part of the recommended criteria. If there is any chance that Opener will potentially need to or benefit from ruffing any of his side-suits, then Zeta should probably not be used because it is impossible for Opener to ascertain anything about Responder's trump holding when Zeta has been used.
♠ KQJ10xxxx, ♥ Ax, ♦ x, ♣AK
This is a decent hand for a Zeta Ask in that it's unlikely that Responder's trump holding will make much difference to trump solidity and it's unlikely that Opener could ever benefit from any ruffs in Responder's hand.
♠ AKQJxxx, ♥ AKxx, ♦ Ax, ♣-
This, on the other hand, is probably not a good hand for Zeta, because if Responder shows up with a holding like Qx or xx in Hearts, Opener will have no means of finding what Responder's trump support is like. Here a Low Beta and then Theta in Spades will almost certainly work out better.
When is a Bid Zeta?
Zeta occurs in two situations only:
A jump-shift by Opener immediately over a positive response to 1♣, and
Complex System only: After the sequence 1♦-1NT(14+)-2NT(4333)-??? now 3-level suit bids are Zeta.
There is no such thing as a repeat Zeta, but high-level Beta is always available if Responder shows a range of controls in response to Zeta.
Responses to Zeta
There are only two scales available for Zeta, depending on whether or not Responder is a passed hand. The scales are identical to those for Beta:
The Normal scale for Zeta is used when Responder's strength is unknown:
: 0-2 Controls
: 3 Controls
: 4 Controls
: 5 Controls
...etc etc (open ended)
The Weak scale for Zeta is used when Responder is a passed hand:
: No Controls
: 1 Control
: 2 Controls
: 3 Controls
: 4 Controls
If Opener does have a fit for Responder's suit, it is often cheaper and easier to use a low-level Beta followed by a Gamma in Responder's suit, even if Opener would rather play in their own suit. They can always jump-shift into their own suit later on, and do have the benefit of being able to make an Epsilon Ask in their own suit, something not possible if they use Zeta. In any case, in such a situation the solidity of the combined fit in Responder's suit will often limit how high the hand can be played in any case. Zeta is most useful when Opener has a singleton or void in Responder's suit, or perhaps a holding such as Ax. If Opener has a holding in responder's suit as good as Hxx it's almost never worth using Zeta.