  We shall show that any complex minimal surface of general type with c_1^2 =
2\chi -1 having non-trivial 2-torsion divisors, where c_1^2 and \chi are the
first Chern number of a surface and the Euler characteristic of the structure
sheaf respectively, has the Euler characteristic \chi not exceeding 4.
Moreover, we shall give a complete description for the surfaces of the case
\chi =4, and prove that the coarse moduli space for surfaces of this case is a
unirational variety of dimension 29. Using the description, we shall also prove
that our surfaces of the case \chi = 4 have non-birational bicanonical maps and
no pencil of curves of genus 2, hence being of so called non-standard case for
the non-birationality of the bicanonical maps.
