  We extend earlier numerical and analytical considerations of the conformally
invariant wave equation on a Schwarzschild background from the case of
spherically symmetric solutions, discussed in Class. Quantum Grav. 34, 045005
(2017), to the case of general, nonsymmetric solutions. A key element of our
approach is the modern standard representation of spacelike infinity as a
cylinder. With a decomposition into spherical harmonics, we reduce the
four-dimensional wave equation to a family of two-dimensional equations. These
equations can be used to study the behaviour at the cylinder, where the
solutions turn out to have logarithmic singularities at infinitely many orders.
We derive regularity conditions that may be imposed on the initial data, in
order to avoid the first singular terms. We then demonstrate that the fully
pseudospectral time evolution scheme can be applied to this problem leading to
a highly accurate numerical reconstruction of the nonsymmetric solutions. We
are particularly interested in the behaviour of the solutions at future null
infinity, and we numerically show that the singularities spread from the
cylinder to null infinity. The observed numerical behaviour is consistent with
similar logarithmic singularities found analytically on the cylinder. Finally,
we demonstrate that even solutions with singularities at low orders can be
obtained with high accuracy by virtue of a coordinate transformation that
converts functions with logarithmic singularities into smooth solutions.
