Case 580C Backhoe starting problems

Sounds like the ignition switch has been installed incorrectly

remove the 4 bolts holding fan to water pump and slip pulley off, replace with new one.

I can talk to you in automotive terms, I know nothing about a backhoe.
I can usually lay the starter motor on the ground, hook up jumper cables properly from starter to battery, at the least, it should kick out the starter drive gear and spin. If it doesn't kick out the drive gear, starter is faulty. I don't know if starter solenoid sits on top of starter motor? I replace as an assembly. Did you check flywheel teeth? I've never worked on a backhoe.

Check the battery and see if it is fully charged. Check your battery cables to see if both end are tight and clean. If corroded needs to be clean and tight. Your starter could be the fault. Start with the battery and cables first then if they are ok check the starter. You can purchase a battery power tester that you can see if power is going to the starter with out tearing things apart.

I am going to have to guess you are referring to a 580 C Case backhoe.
In any case never ever put anything in a brake fluid reservoir but brake fluid, and the right fluid to be sure, or you will seriously damage or destroy the brakes "forever".
The brake fluid reservoir is usually plastic, and often you can see through it just enough to see the level of brake fluid.
It should be located near the front of the brake pedal, and is usually sort of white in color, with a black cap that can be screwed off.
God bless your efforts.

Valves not sealing in the remote control block could be the problem or ram seals.

there will be small seals or "O" rings in the pump shaft cover. Manual details will be on the net but best if you have a case technician repair it as it will be done quickly and properly.

It sounds like you have a bad starter. You can try bringing 12v direct to
the starter just to be sure(be sure it's in neutral) or use a volt meter at
the starter while somone turns the key. You have checked the wire's
and all appears to be ok, the only thing left that would draw the meter
into the red is the starter itself. I hope this helps.

check this link :
http://www.ioffer.com/i/CASE-580C-580-C-LOADER-BACKHOE-TRACTOR-SERVICE-MANUAL-117272901

Pregunta , tienes, el manual de la case 580 ck, donde se muestra el circuito hidrailico de la transmision, si es asi, embiame una foto del circuito o el manual y te puedo apoyar.