You can certainly have a machine in the 10.37 which did not use the Cisco as its default gateway, but it could cause problems reaching all the machines in the local network. It really depends on how this network is set up. If it is set up with internal
subnets or VLANs, all machines really need to use the Cisco as their default gateway.
If you want a machine to be reachable from the Internet, there is really no option. Its default gateway must be an Internet router. You can use a static route to redirect particular traffic (such as traffic from a particular site or subnet) to
use an alternative gateway. But to be reachable from any possible user of the Internet, all traffic must use that gateway, so it needs to be a default route.
In a simple network, where all machines are in the same IP subnet and the only router is a gateway or Internet router, things are simple. If your local network has an internal router which controls VLANs, things are much more complicated.
I would look at the problem in a different way. I would configure this server in its own network with its own IP subnet behind its own Internet router, using this router as its default gateway and get that working. I would then consider how I could
route between this machine and the 10.37 network using static routing (and adding the LAN router) so that it can communicate with machines on your local LAN. A simple diagram would look like this.
Internet
|
DIR-100
192.168.3.1
|
Application server
Remote access
192.168.3.x
dg 192.168.3.1
|
|
192.168.3.254 dg blank
LAN router
10.37.2.93 dg blank
Then configure static routes in both subnets so that all traffic from 192.168.3.0 can reach 10.37.0.0 and all traffic from 10.37.0.0 can reach 192.168.3.0 through your LAN router (which could be a hardware device or a RRAS server).