Partial decidable presentations in hyperarithmetic

We study the problem of the existence of decidable and positive $\Pi_1^1$- and $\Sigma_1^1$-numberings of the families of $\Pi_1^1$- and $\Sigma_1^1$-cones with respect to inclusion. Some laws are found that reflect the presence of decidable computable $\Pi_1^1$- and $\Sigma_1^1$-numberings of these families in dependence on the analytical complexity of the set defining a cone.