I resized your photos and pasted them here to make them easier for people to look at..
Looks as if you have a Peerless Transmission.
Cant say 100% for sure. But it looks to me as if you have bad oil seals.
Clean the area real good. Then "drive" the mower for a few minutes, stopping very few seconds and look for oil coming from around the seals or from between the adapter and transmission case if you can not see the seal.
If you find that it is the seals and have or replace them yourself. Try to keep grass blades, string and other items from wrapping around the output shafts behind the axle adapters. For grass and such creates a lot of heat and can shorten an oil seals life a great deal.
It's a good Idea to check it every year, and fill /replace the grease as needed. Peerless use Bentonite grease. Think it holds 14oz., which is how it comes. j-thomas.com has it for 6.99.
(I use a Lithium based grease myself, but that's beside the point) http://j-thomas.com/Catalog/211.htm
Thanks for letting Hopper know that the transmission does not have seals.
I know that the one on my Snapper walk behind does not have them and was not sure if that applied to all Peerless tansmissions or not.
The trans does not have a drain on it. The grease is too thick to run out. You will have to disassemble the top and bottom case halves then wash all the old grease out with solvent. Then squeeze the grease out of the tube into the trans and reassemble. It is REAL MESSY!!!!!