  The appearance of eruptive space plasma processes, e.g., in eruptive flares
as observed in the solar atmosphere, is usually assumed to be caused by
magnetic reconnection. The process of magnetic reconnection is often connected
with singular points of the magnetic field. We therefore analyse the system of
stationary resistive/non-ideal magnetohydrodynamics (MHD) in the vicinity of
singular points of flow and field to determine the boundary between
reconnection solutions and non-reconnective solutions. We find conditions to
enable the plasma to cross the magnetic separatrices also inside the current
sheet, close to the current maximum. The results provide us with the
topological and geometrical skeleton of the resistive MHD fields. We therefore
have to perform a local analysis of almost all non-ideal MHD solutions with a
generalized non-idealness. We use Taylor expansions of the magnetic field, the
velocity field and all other physical quantities, including the non-idealness,
and with the method of a comparison of the coefficients, the non-linear
resistive MHD system is solved analytically. It turns out that not every
non-ideal flow is a reconnective flow and that pure resistive/non-ideal MHD
only allows for reconnection-like solutions, even if the non-idealness is
localized to the region around the magnetic null point. It is necessary that
the flow close to the magnetic X-point is also of X-point type to guarantee
positive dissipation of energy and annihilation of magnetic flux. If the
non-idealness has only a one-dimensional, sheet-like structure, only one
separatrix line can be crossed by the plasma flow, similar to reconnective
annihilation solutions.
