  A construction of the Virasoro algebra in terms of free massless
two-dimensional boson fields is studied. The ansatz for the Virasoro field
contains the most general unitary scaling dimension 2 expression built from
vertex operators. The ansatz leads in a natural way to a concept of a quasi
root system. This is a new notion generalizing the notion of a root system in
the theory of Lie algebras. We introduce a definition of a quasi root system
and provide an extensive list of examples. Explicit solutions of the ansatz are
presented for a range of quasi root systems.
