  Toda conformal field theories are natural generalizations of Liouville
conformal field theory that enjoy an enhanced level of symmetry. In Toda
conformal field theories this higher-spin symmetry can be made explicit, thanks
to a path integral formulation of the model based on a Lie algebra structure.
The purpose of the present document is to explain how this higher level of
symmetry can manifest itself within the rigorous probabilistic framework
introduced by R. Rhodes, V. Vargas and the first author. One of its features is
the existence of holomorphic currents that are introduced via a rigorous
derivation of the Miura transformation. More precisely, we prove that the
spin-three Ward identities, that encode higher-spin symmetry, hold in the
$\mathfrak{sl}_3$ Toda conformal field theory; as an original input we provide
explicit expressions for the descendent fields which were left unidentified in
the physics literature. This representation of the descendent fields provides a
new systematic method to find the degenerate fields of the $\mathfrak{sl}_3$
Toda (and Liouville) conformal field theory, which in turn implies that certain
four-point correlation functions are solutions of an hypergeometric
differential equation of the third order.
