  In this paper we develop the generalised Schur theory offered in the recent
paper by the second author in dimension one case, and apply it to obtain two
applications in different directions of algebra/algebraic geometry.
  The first application is a new explicit parametrisation of torsion free rank
one sheaves on projective irreducible curves with vanishing cohomology groups.
  The second application is a commutativity criterion for operators in the Weyl
algebra or, more generally, in the ring of ordinary differential operators,
which we prove in the case when operators have a normal form with the
restriction top line (for details see Introduction).
  Both applications are obtained with the help of normal forms. Namely,
considering the ring of ordinary differential operators $D_1=K[[x]][\partial ]$
as a subring of a certain complete non-commutative ring $\hat{D}_1^{sym}$, the
normal forms of differential operators mentioned here are obtained after
conjugation by some invertible operator ("Schur operator"), calculated using
one of the operators in a ring. Normal forms of commuting operators are
polynomials with constant coefficients in the differentiation, integration and
shift operators, which have a restricted finite order in each variable, and can
be effectively calculated for any given commuting operators.
