  Matrix elements in different representations are connected by quadratic
relations. If matrix elements are those of a $\textit{group element}$, i.e.
satisfying the property $\Delta(X) = X\otimes X$, then their generating
functions obey bilinear Hirota equations and hence are named $\tau$-functions.
However, dealing with group elements is not always easy, especially for
non-commutative algebras of functions, and this slows down the development of
$\tau$-function theory and the study of integrability properties of
non-perturbative functional integrals. A simple way out is to use arbitrary
elements of the universal enveloping algebra, and not just the group elements.
Then the Hirota equations appear to interrelate a whole system of generating
functions, which one may call $\textit{generalized}$ $\tau$-functions. It was
recently demonstrated that this idea can be applicable even to a somewhat
sophisticated case of the quantum toroidal algebra. We consider a number of
simpler examples, including ordinary and quantum groups, to explain how the
method works and what kind of solutions one can obtain.
