The generalisation involves allowing different values of the exponent , [math]x[/math].
The "generalised Lorentz equation" appears in the relativistic ellipse derivation of an apparent spectrum of possible relativistic theories, which includes special relativity and discredited C19th ballistic emission theory.
While the standard Lorentz equation is considered fundamental because it can be used in transforms that keep certain critical properties unchanged, those properties are presumably also unchanged if the same transform is applied two, three, or more times, producing a sequence of candidate theories at half-integer spacings.
If an equation used in a transform preserves a quantity, then this at least suggests the possibility that the same preservation might also hold for simple fractional powers, in which case we would effectively have another infinite number of exponent values with the same preservation property.
In the relativistic ellipse exercise, the range of potential solutions is in fact continuous, with each solution related to every other solution by a suitable Lorentzlike factor.
Main candidate theories
If we take the Dopppler equations associated with simple stationary flat spacetime as our reference, special relativity's relationships for visible frequency and length are "redder and shorter" by a single Lorentz factor (x=0.5), and the relationships required by a relativistic acoustic metric are in turn even "redder and shorter" than those of SR, by an additional Lorentz factor (x=1).
A range of intermediate theories are also conceivable but fail to either be compatible with flat spacetime (SR), or with quantum mechanics (super-GR)) are perhaps currently difficult currently difficult to justify.