I am a fellow owner of an E35, but I'm not sure that I can solve your problem. Please clarify what is happening, however...
What I'm gathering is that you can get the engine to start and run for a few seconds, but it dies soon after starting and will not stay running.

The starting procedure is to turn the operating switch (mounted on the handle) to "on". Pull out the choke lever, and pull up the throttle to the fast postition. Pull the starter cord, and once the engine starts, push the choke lever in.

If this is what you are doing and it won't run correctly, I would remove the spark plug and see if you get spark when you pull the engine over (with the switch in the on position). Keep the threaded end of the spark plug in contact with a metal part of the engine housing for grounding to see if you've got spark. If you don't have spark, try a different plug, then suspect a potentially bad coil if you don't get spark with the second plug.
If you do have spark, another possible cause is the carburetor / fuel delivery. I have had trouble starting mine if I let the fuel level get low in the tank. The fuel tank is large on that engine, but there is still a fair amount of gasoline in mine when it runs out of gas. I would guess that there is still 1/2" or more of gasoline depth in the bottom of the tank when mine runs out of gas. If you've got plenty of gas in the tank, the carburetor may need to be cleaned.

Hopefully you can get yours running correctly. That model of edger is one of the finest walk-behind units ever made.

Sir,
I thnak you for your help and will try your instructions. I did not get an owners manual so your starting instructions will be most helpful.
Thank you

The problem I had with the same edger and same symptoms is a rusty gas tank. I re coated mine using a motorcycle POR-15 gastank kit. It now works fine. A new tank will cost $90. The kit was $40. You can also use The Works toilet bowl cleaner in it to remove the rust and keep the tank full after each use. That will slow down future rust. I also had to clean all the ports on the carb with carb cleaner in a spray can using the straw. I completely took mine apart. That is every bolt and nut, to re paint it. I put a new belt on it and re assembled it and it looks and runs like new. It is the best edger you can find. They don't make them anymore. Now your best bet is the junk at sears or a stick edger. Good Luck.

Great edger. Too bad it's not made anymore. Most E35's came with 3hp or 3.5hp Briggs, but I understand some came with Kawasaki engines. Which engine is on your edger? Will help if you post engine model and type numbers.

As for comment that only "junk at sears" is only option is apparently mis-informed. Deere dropped the E35 because they weren't competitive with high end edgers like the Power Trim and Little Wonder, both of which are considered the Gold Standard with professional landscapers.

I'm looking for a fuel filter for my E35 Edger. Can anybody help me.

This is the carb linkage intact. 'Took me awhile to figure how it went back together. The gas tank is still clogged tho. Thanks for the tip above.