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This is the first of at least two articles that describe the moduli spaces ofpseudoholomorphic, multiply punctured spheres in R x (S^1 x S^2) as defined bya certain natural pair of almost complex structure and symplectic form. Thisarticle proves that all moduli space components are smooth manifolds. Necessaryand sufficient conditions are also given for a collection of closed curves inS^1 x S^2 to appear as the set of |s| --> infinity limits of the constant s inR slices of a pseudoholomorphic, multiply punctured sphere.

Pseudoholomorphic punctured spheres in R x (S^1 x S^2) : Properties and existence