You have a mask of 255.255.255.255
on the top two rows, which seems a little odd to me.
So are your subnets:-
- 10.10.126.0-63 routing via 10.10.136.2 on eth0
- 10.6.0.0-63.......no routing from eth2
It would suggest that if you should be able to ping addresses 10.6.0.0-63 from your server but not beyond. Is there a route via 10.10.136.2 back across to 10.6..x.x? If not then your eth2 does not know how to respond.
I'm on RHEL 6.3 (you don't say, or I've missed it) but have a look in /etc/sysconfig/network-scripts/ifcfg-eth0 and eth1 if you have them.
Be careful if you make backup copies in the same directory as the process to set them up at boot time will read all files called ifcfg-*
We found this out by 'experience', i.e. a big mess that took us a while to work out.