  In a 1979 paper, K. Okamoto introduced the space of initial values for the
six Painlev\'e equations and their associated Hamiltonian systems, showing that
these define regular initial value problems at every point of an augmented
phase space, a rational surface with certain exceptional divisors removed. We
show that the construction of the space of initial values remains meaningful
for certain classes of second-order complex differential equations, and more
generally, Hamiltonian systems, where all movable singularities of all their
solutions are algebraic poles (by some authors denoted the quasi-Painlev\'e
property), which is a generalisation of the Painlev\'e property. The difference
here is that the initial value problems obtained in the extended phase space
become regular only after an additional change of dependent and independent
variables. Constructing the analogue of space of initial values for these
equations in this way also serves as an algorithm to single out, from a given
class of equations or system of equations, those equations which are free from
movable logarithmic branch points.
