  Painleve transcendents are usually considered as complex functions of a
complex variable, but in applications it is often the real cases that are of
interest. Under a reasonable assumption (concerning the behavior of a dynamical
system associated with Painleve IV, as discussed in a recent paper), we give a
number of results towards a classification of the real solutions of Painleve IV
(or, more precisely, symmetric Painleve IV), according to their asymptotic
behavior and singularities. We show the existence of globally nonsingular real
solutions of symmetric Painleve IV for arbitrary nonzero values of the
parameters, with the dimension of the space of such solutions and their
asymptotics depending on the signs of the parameters. We show that for a
generic choice of the parameters, there exists a unique finite sequence of
singularities for which symmetric Painleve IV has a two-parameter family of
solutions with this singularity sequence. There also exist solutions with
singly infinite and doubly infinite sequences of singularities, and we identify
which such sequences are possible (assuming generic asymptotics in the case of
a singly infinite sequence). Most (but not all) of the special solutions of
Painleve IV correspond to nongeneric values of the parameters, but we mention
some results for these solutions as well.
