  Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is
a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i
belongs to S for all i, and none of the points w_i, i>0, lie on the half-line
H= {(k,0): k =< 0}.
  In a recent paper, G. Schaeffer and the author computed the length generating
function S(t) of walks on the slit plane for several sets S. All the generating
functions thus obtained turned out to be algebraic: for instance, on the
ordinary square lattice,
  S(t) =\frac{(1+\sqrt{1+4t})^{1/2}(1+\sqrt{1-4t})^{1/2}}{2(1-4t)^{3/4}}.
  The combinatorial reasons for this algebraicity remain obscure.
  In this paper, we present two new approaches for solving slit plane models.
One of them simplifies and extends the functional equation approach of the
original paper. The other one is inspired by an argument of Lawler; it is more
combinatorial, and explains the algebraicity of the product of three series
related to the model. It can also be seen as an extension of the classical
cycle lemma. Both methods work for any set of steps S.
  We exhibit a large family of sets S for which the generating function of
walks on the slit plane is algebraic, and another family for which it is
neither algebraic, nor even D-finite. These examples give a hint at where the
border between algebraicity and transcendence lies, and calls for a complete
classification of the sets S.
