  We have completed part of a program to study the X-ray emission properties of
old rotation-powered pulsars with XMM-Newton in order to probe and identify the
origin of their X-radiation. The X-ray emission from these old pulsars is
largely dominated by non-thermal processes. None of the observed spectra
required adding a thermal component consisting of either a hot polar cap or
surface cooling emission to model the data. The X-ray spectrum of PSR 0950+08
is best described by a single power law of photon-index
1.93^{+0.14}_{-0.12}.Taking optical data from the VLT FORS1 into account a
broken power law model is found to describe the pulsar's broadband spectrum
from the optical to the X-ray band. Temperature upper limits for possible
contributions from a heated polar cap or the whole neutron star surface are
T_{pc} < 0.87 x10^6 K and T_s < 0.48 x 10^6 K, respectively. We also find that
the X-ray emission from PSR 0950+08 is pulsed with two peaks per rotation
period. The phase separation between the two X-ray peaks is ~144 degree. The
main radio peak and the trailing X-ray peak are almost phase aligned. The
fraction of X-ray pulsed photons is ~30%. A phase-resolved spectral analysis
confirms the non-thermal nature of the pulsed emission. Detailed pulse profile
simulations constrain the pulsar's emission geometry to be that of an almost
orthogonal rotator. The spectral emission properties observed for PSR 0823+26
are similar to those of PSR 0950+08. For PSR J2043+2740 we report the first
detection of X-ray emission. A power law spectrum, or a combination of a
thermal and a power law spectrum all yield acceptable descriptions of its X-ray
spectrum. No X-ray pulses are detected from PSR J2043+2740. A pulsed fraction
upper limit is 57%.
