  It is known that $Q$-conditional symmetries of the classical Burgers'
equation express in terms of three functions satisfying a coupled system of
Burgers-like equations. The search of conditional symmetries of this system
leads to a system of five coupled Burgers-like equations. Iterating the
procedure, an infinite hierarchy of systems made of an odd number of coupled
Burgers-like equations is obtained. Moreover, starting from a pair of
Burgers-like equations, a similar hierarchy of systems made of an even number
of coupled Burgers-like equations arises; it is also proved that the two
hierarchies may be unified. Writing a generic element of this hierarchy as a
matrix Burgers' equation, the existence of the matrix Hopf-Cole transformation
allows for its linearization and the determination of its solution. Finally, it
is shown that each element of the hierarchy possesses a five-dimensional Lie
algebra of classical point symmetries. Though these Lie algebras are realized
in manifolds with different dimensionality, they are all isomorphic.
