  We show that the algebraic dimension of a twistor space over n#CP^2 cannot be
two if n>4 and the fundamental system (i.e. the linear system associated to the
half-anti-canonical bundle, which is available on any twistor space) is a
pencil. This means that if the algebraic dimension of a twistor space on
n#CP^2, n>4, is two, then the fundamental system either is empty or consists of
a single member. The existence problem for a twistor space on n#CP^2 with
algebraic dimension two is open for n>4.
