  We derive the equations corresponding to twisting type-N vacuum gravitational
fields with one Killing vector and one homothetic Killing vector by using the
same approach as that developed by one of us in order to treat the case with
two non-commuting Killing vectors. We study the case when the homothetic
parameter $\phi$ takes the value -1, which is shown to admit a reduction to a
third-order real ordinary differential equation for this problem, similar to
that previously obtained by one of us when two Killing vectors are present.
