  We present a complete solution to the problem of Formal Higher Spin Gravities
--- formally consistent field equations that gauge a given higher spin algebra
and describe free higher spin fields upon linearization. The problem is shown
to be equivalent to constructing a certain deformation of the higher spin
algebra as an associative algebra. Given this deformation, all interaction
vertices are explicitly constructed. All formal solutions of the equations are
explicitly described in terms of an auxiliary Lax pair, the deformation
parameter playing the role of the spectral one. We also discuss a natural set
of observables associated to such theories, including the holographic
correlation functions. As an application, we give another form of the Type-B
formal Higher Spin Gravity and discuss a number of systems in five dimensions.
