  We introduce W-spin structures on a Riemann surface and give a precise
definition to the corresponding W-spin equations for any quasi-homogeneous
polynomial W. Then, we construct examples of nonzero solutions of spin
equations in the presence of Ramond marked points. The main result of the paper
is a compactness theorem for the moduli space of the solutions of W-spin
equations when W is a non-degenerate, quasi-homogeneous polynomial whose
variables all have weight (or fractional degree) wt(x_i) < 1/2. In particular,
the compactness theorem holds for the A,D, and E superpotentials.
