никакой маршрут к шлюзу по умолчанию

Firewall on the router was blocking access to port :53 Added

-A RH-Firewall-1-INPUT -p udp -m state --state NEW -m udp --dport 53 -j ACCEPT
-A RH-Firewall-1-INPUT -p tcp -m state --state NEW -m tcp --dport 53 -j ACCEPT

Sorry for the hassle (and thanks sejo for helping out !)

edit you interfaces to make it look something like this

sudo vi /etc/network/interfaces

auto lo
iface lo inet loopback

# The primary network interface
auto eth0
iface eth0 inet static
    address 10.1.1.10
    netmask 255.255.255.0
    network 10.1.1.0
    broadcast 10.1.1.255
    gateway 10.1.1.1
    # dns-* options are implemented by the resolvconf package, if installed
    dns-nameservers 10.1.1.100
    dns-search test.net
    dns-domain test.net

You'll most probably want to change your dns-nameserver - if you don't know what that is, just use 8.8.8.8. I would assume your IP settings are ok so no need to change them. Also update the dns-search and dns-domain and append your internal domain name. If you update your resolv.conf file, it may get over ridden when your network services restart as the resolvconf package will take what is placed in your interfaces file and put them in to your resolv.conf file. Thus the problem may reappear. Also the routers firewall may be blocking ports or not serving DNS. Which is why you could try 8.8.8.8. This will give you external DNS resolution. If internal DNS isn't working, then you need to ensure a DNS server is contactable and running. Otherwise use the /etc/hosts file.