Generalized computable numberings and fixed points

The talk considers generalized computable numberings from the point of view of uniform enumerability of numbered families relative to arbitrary oracles. The results presented are aimed at classifying oracles such that all (principal) families computable in them have generalized computable numberings that satisfy the Kleene fixed point theorem with different degrees of uniformity: complete and precomplete numberings, weakly precomplete numberings, and also numberings that satisfy the Recursion theorem and the Recursion theorem with parameters.