  Data accumulated recently for the exclusive measurement of the $pp\to
pp\pi^+\pi^-$ reaction at a beam energy of 0.793 GeV using the COSY-TOF
spectrometer have been analyzed with respect to possible events from the $pp
\to nn\pi^+\pi^+$ reaction channel. The latter is expected to be the only
$\pi\pi$ production channel, which contains no major contributions from
resonance excitation close to threshold and hence should be a good testing
ground for chiral dynamics in the $\pi\pi$ production process. No single event
has been found, which meets all conditions for being a candidate for the $pp
\to nn \pi^+\pi^+$ reaction. This gives an upper limit for the cross section of
0.16 $\mu$b (90% C.L.), which is more than an order of magnitude smaller than
the cross sections of the other two-pion production channels at the same
incident energy.
