is it possible to list LAN names when DNS is managed by ext router?

I wish to know the computer name associated with a known ip-address. The DNS is managed by my router (WIND3 webcube) and I can see the resolved name using the browser (http://192.168.1.1) Nevertheless, I wish to read the LAN computer names by command line from the console. I've tried <ping -r 192.168.1.255> then <arp -a> as suggested by Bodo; this has shown all IPs & relevant MACs on the LAN except my own; the only hostname shown has been the router name corresponding to 192.168.1.1 - half question has been answered, now I have a way to discover remote IPs on the LAN by console. In addition, using <nmap -sn 192.168.1.0/24>, I can see the full list of IPs on the LAN but still no hostnmes.

At the moment the best solution is running this script:
for x in {101..110}
do nslookup 192.168.1.$x | grep name
done

that shows all active names and relevant IPs (reversed)

workarounds(using ping, arp table and scans) that should work in most networks but might not report all devices(reports only those devices that echo/respond to the ping).

With ping and arp

ping(for a short while) the broadcast IP address(i.e. the highest, last and unusable IP address in your sub-net e.g. 192.168.1.255) which will in turn result in ping packets being sent to every possible IP address on your network like so:

ping -b 192.168.1.255

Then, run an arp(Address Resolution Protocol) request like so:

arp -a

With nmap

From man nmap:

-sn (No port scan)
  This option tells Nmap not to do a port scan after host discovery,
  and only print out the available hosts that responded to the host
  discovery probes. This is often known as a “ping scan”.

So, you can use it on your network like so:

nmap -sn 192.168.1.0/24

Where 192.168.1.0(the first unusable IP address i.e. 0) is your network address and 24 is your sub-net mask.

To get the name for a single known IP address from the router you can use

dig @router -x ipaddress +short

for example

dig @192.168.1.1 -x 192.168.1.100 +short

see man dig

I was not able to use dig's query type axfr or nslookup's ls command to get a full listing from my router. This might be a permissions issue.

Of course you can use shell programming to run dig in a loop for a range of IP addresses.