Paul Davies, Professor of Natural Philosophy at the University of
Adelaide, wrote (in his book "Are We Alone?"), "The laws of physics
(with a minor and irrelevant exception) seem to be symmetric as far as
past and future is concerned." Can anyone tell me about this "minor
and irrelevant" exception, which seems to imply that, under certain
circumstances, time can travel backwards?

Perhaps he was referring to tachyon theory:

http://en.wikipedia.org/wiki/Tachyon

http://plato.stanford.edu/entries/causation-backwards/#4.2

I assume Davies is talking about the violation of time reversal
invariance that occurs in certain exotic weak interactions. See, for
example:

http://www2.slac.stanford.edu/vvc/theory/timereversal.html

While this asymmetry does occur, the interactions are the same if you
reverse time, flip the interaction over in a mirror, and change all
particles into their antiparticles. This is the famous CPT invariance,
where the initials stand for charge (particle/antiparticle), parity
(left/right), and time (forward/backward). The violation of parity
symmetry was completely unexpected and won a Nobel prize. It has been
demonstrated that CP (and therefore T) symmetry is violated. For
theoretical reasons, it would be very hard to believe that CPT is
violated.

Davis is talking about  Experiment with Kaons

http://hyperphysics.phy-astr.gsu.edu/hbase/particles/cronin.html

it nothing to do with time travel.

And I just want to add to your previous question:
        is there possibly a connection 'dark matter' and ether

that if I would think that
   "question doesn't make sense"
I would not answer it. I would either leave it alone or ask you to
clarify it.  Ether had very strange properties: to get the very high
speed for light, it woud have to be very rigid - and yet, bodies could
pass through it. 'Dark Matter' as well as 'Dark Energy' also have some
strange properties. I pointed out in my answer that both problems may
have the same resolution. 
While I do see some problems with the 'dark matter' hypothesis, I have
said it is an open question: jury is still out. In my view, a person
who does not have some doubts about open questions in science, is not qualified
to answer questions form the public. 
Hedgie

a CLOSED TIMELIKE LOOP