  Fefferman-Graham ambient construction can be formulated as
$\mathfrak{sp}(2)$-algebra relations on three Hamiltonian constraint functions
on ambient space. This formulation admits a simple extension that leads to
higher-spin fields, both conformal gauge fields and usual massless fields on
anti-de Sitter spacetime. For the bulk version of the system, we study its
possible on-shell version which is formally consistent and reproduces conformal
higher-spin fields on the boundary. Interpretation of the proposed on-shell
version crucially depends on the choice of the functional class. Although the
choice leading to fully interacting higher-spin theory in the bulk is not
known, we demonstrate that the system has a vacuum solution describing general
higher-spin flat backgrounds. Moreover, we propose a functional class such that
the system describes propagation of higher-spin fields over any higher-spin
flat background, reproducing all the structures that determine the known
nonlinear higher-spin equations.
