  The paper deals with first order self-adjoint elliptic differential operators
on a smooth compact oriented surface with non-empty boundary. We consider such
operators with self-adjoint local boundary conditions. The paper is focused on
paths in the space of such operators connecting two operators conjugated by a
unitary automorphism. The first result is the computation of the spectral flow
for such paths in terms of the topological data over the boundary. The second
result is the universality of the spectral flow: we show that the spectral flow
is a universal additive invariant for such paths, if the vanishing on paths of
invertible operators is required.
  In the next paper of the series we generalize these results to families of
such operators parametrized by points of an arbitrary compact space instead of
an interval. The integer-valued spectral flow is replaced then by the family
index taking values in the $K^1$-group of the base space.
