I have a domain name and would like to let another dns server handle all third-level domain names.

For instance, abc.com's DNS servers are:

ns1.xxx.com
ns2.xxx.com

I'd like all subdomains in the form of SUBDOMAIN.a.abc.com to be handled by these DNS servers:

ns1.zzz.com
ns2.zzz.com

Is this possible to do? How do I go about configuring the initial ns1.xxx.com so that it will let another dns server handle all domain resolutions for addresses in the "a.abc.com" subdomain?

a.abc.com. 14400 IN NS ns1.zzz.com.
a.abc.com. 14400 IN NS ns2.zzz.com.

would be for BIND - add both lines to your zone file
for abc.com and then a.abc.com will be delegated to
the nameservers ns{1,2}.zzz.com, after reload.