[complete rewite]
If I under stand it correctly you have an intranet with 10.0.0.0/8 as numbering scheme.
(10.0.0.0 - 10.255.255.255, or up to 16777214 different addresses)
Your desktop has two network cards. One of them is connected to that intranet and it has the IP 10.35.134.1.
You also have a network with access to the Internet. That network is on 192.168.1.0/24 ( 192.168.1.0 - 192.1658.0.255, or up to 254 different addresses on this network).
Your desktops second NIC is connected to this network.
Since your desktop is multi-homed you will need to specify which network to use in the routing tables. You can set one (and only one) of them to the default for routes which are not specified.*1
Since you do not want to add routes to every single network on the Internet you want to use the gateway on that network as the default route.
This seems to match your
Internet settings are
192.168.1.15 The NICs IP
255.255.255.0 The netmask for that net (a /24)
192.168.1.1
I am guessing that 192.168.1.1 is the default gateway on that network.
However the other triplet confuses me.
10.35.134.129
255.0.0.0
10.35.134.1
I understand the IP and mask for the intranet LAN. But the third value confuses me.
The only things I can guess at is that you have configured one or both of these NICs via DHCP, that you set the DG via DHCP and that the DHCP lease time for one of the networks is longer than that from the other, overwriting part of your settings when it is renewed.
Granted, that includes quite a few assumptions.
*1 Technically not quite true. You can use multiple default routes. But only one will be used at any time.