  We continue the enumeration of plane lattice paths avoiding the negative
quadrant initiated by the first author in [Bousquet-M{\'e}lou, 2016]. We solve
in detail a new case, the king walks, where all 8 nearest neighbour steps are
allowed. As in the two cases solved in [Bousquet-M{\'e}lou, 2016], the
associated generating function is proved to differ from a simple, explicit
D-finite series (related to the enumeration of walks confined to the first
quadrant) by an algebraic one. The principle of the approach is the same as in
[Bousquet-M{\'e}lou, 2016], but challenging theoretical and computational
difficulties arise as we now handle algebraic series of larger degree. We also
explain why we expect the observed algebraicity phenomenon to persist for 4
more models, for which the quadrant problem is solvable using the reflection
principle.
