The A10 has an independent routing table for the management interface that doesn't interfere to the data routing table of 'shared' partition or any data routing table of any private partition. You can confirm that by issuing in the 'shared' partition the commands 'show ip route' and 'show ip route mgmt', and you'll see how the two different routing tables look like. You can configure the mgmt routing table default gateway with the 'ip default-gateway' command under the management interface configuration mode, where you do that in the data routing table with the 'ip route 0.0.0.0/0' command (assuming you want to configure a static default gateway there).
Comments
I hope this helps.
Daniel.