Wristpin

By the look of the image your machine has points ignition, no trigger module between the left leg of the coil armature and the coil itself.
Plug caps. Comments have been made about these being the likely cause of failure. This is only likely to be the case if the original BS connector has been removed and replaced with an aftermarket cap. Again, from your image you appear to have the original connector and there's not a lot that can go wrong with those. Sometimes they have a simple rubber boot. 
The only exterior things left to check is the kill switch on the throttle plate or the kill wire that goes to it. If all is OK there, you have a choice, either remove  the flywheel and examine the points etc , or find a later BS Magnetron coil ( with suitable leg spacing)  and fit that in place of yours - a ten minute job and a direct replacement using your existing flywheel without the need to remove it.
Personally I would do the latter as they are 99.9% reliable and obviate  the need to ever remove the flywheel again.
 

Pity about the box, people will leave them out side to fill with water. I've reskinned a few with "tin" from 25ltre oil drums, the trick is finding a drum that's nor pre-dented. The 17" Super punch was a good machine and a perfectly useable Classic now: as is the 14" MkVII Punch .Here's a few of mine , the one on the right is the quite uncommon ten bladed "Professional" .
 

The 17" series 1 Super Punch ran from 1956 to 1959 and the same basic design carried on into the 80s badged as a Qualcast.
The screw height adjuster , " kidney" fuel tank  and 75cc on the engine plate would suggest 1963 onwards but if its original engine was 98cc  it would be considerably later, probably around 1967 .

Pity about the box, people will leave them out side to fill with water. I've reskinned a few with "tin" from 25ltre oil drums, the trick is finding a drum that's nor pre-dented. The 17" Super punch was a good machine and a perfectly useable Classic now: as is the 14" MkVII Punch .Here's a few of mine , the one on the right is the quite uncommon ten bladed "Professional" .
 

A properly set up diaphragm will help but do observe the choke butterfly and see that it is closing correctly. The butterfly spindle is in two separate parts that push in and clip to the butterfly flap. Those spindle sections eventually wear oval and the flap no longer seals. Assuming that they are still available its a two minute job to change them. The choke flaps came in three types, plain with no hole, with an open hole and with a hole with a spring poppet valve. If you have a bad starter it may be worth fitting a plain flap or sealing the hole in the existing one if applicable.
It may seem obvious but those models with the fuel tank below the carb always need a few pulls to lift the fuel but a full tank helps. Also they often cold start better with the throttle just off the stop position and the engine allowed to warm up and settle for a few seconds before fully opening the throttle slowly. 

What a coincidence, I've just dug out one of that last Rotoscythes made under the Webb Windsor banner that I've had in store for 30+ years . Despite main bearing wear that makes setting the points interesting it makes a good job of the lawn.
https://www.dropbox.com/s/9zrlm7flb2hg0xv/IMG_1989.MOV?dl=0

Standard old school practice but not with petrol  - elf and safety. Paraffin , diesel or even water, but if valves are correctly lapped they will seal correctly. Only ever use the paste from the fine end of the tin; if that is not sufficient they need to be refaced and the seats cut. If the coarse paste is needed to recover a bad valve or seat the chances are that you will end up with a convex valve face and cupped seat. 
The days when cars needed a decoke every ten thousand miles and every village garage had refacing and seat cutting facilities are gone - as are the garages: so us small engine types have to seek out those who have the kit. A decent garden machinery workshop should have it as will vintage motor bike shops but otherwise it will be engine reconditoners who may turn up their noses at our sort of kit.

Certainly good valve sealing and adjustment is the basis of a sweet running engine. Don't just do the exhaust, I've found that the thirty degree inlet vales on 5hp BS engines seem to suffer a bit and benefit as much from a re- seat  as much as the supposedly harder worked exhaust.
Quite often poor starting on 21s and Ospreys was down to nothing more technical than throttle cable adjustment not applying the choke  fully. Back in the day it did seem that the standard fit throttle control only had just enough throw to give both full choke and stop. Couple that with a bit of wear and slop in the pivot of the lever on the carburettor throttle plate and it was a choice of one ore the other but not both. We would often set them up for a reliable full choke and fit a handle bar mounted kill switch to look after stopping.

Very odd - short of spinach?  I fitted lots of Sprints over the years and never experienced any issues  - just a good steady pull with no snatching. Even the Professionals with the 5HP BS require just a steady pull . I've just sold a tidy 3.5hp BS one dated 1969 with points ignition and  the fuel tank below the carb - from standing over night, two slow pulls with the ignition killed to lift the fuel, and then one on 1/4 throttle and away she went. I find that the cutting disc works for you rather than against you.

Yes, the valve clearances are "adjustable"  - if too tight a small amount may be ground off the end of the valve stem to achieve the book clearance. This is done after any valve refacing, seat cutting and lapping etc.
Loss of spark may well be down to a faulty coil or condenser, but first check the contact breaker points gap and and also that the breaker plunger is not sticking in the block when warmed up. If that happens the points are held open until things cool down.
I've found an old micro fiche covering engines 142300 - 142457 and the relevant parts are shown as-
Coil 296834, condenser 29861, contact breaker set 29667, breaker plunger 65704
If you are not concerned about "originality" and the coil is good, the points and condenser can be replaced with a solid state trigger module such as a Meco or a Nova. If the coil is kaput and unavailable and the engine has electric start , there is a "work around" using an external car type coil as used on some Kohlers.  

...... or the condenser or even tight valve clearances.
 
 
 
 
 

The Meco electronic trigger  works fine with Japs, BSAs and Villiers provided that they do not have the later coils incorporating an embedded condenser. Just discard the points and condenser and connect the Meco to the coil LT. I fit them as standard when refurbishing Slopers on Marquises.
 

Looks as though it could br a combined centrifugal clutch and vari-speed pulley . There would be some sort of mechanism to take up the slack or apply tension to the belt to alter its position in the pulley to effect the change of speed, possibly controlled by the heavy cable and lever on the left hand handle bar. An image of what ever is under the guard on the left may give a clue. 

DR was a US brand that appeared in the UK in the late 80s , I think. I went to a dem somewhere in West Sussex and was impressed enough to buy half a dozen . Unfortunately the punters didn't share my entheusiasm and the last one stayed on our books as a hire machine. They had a heavy duty nylon line head which took 4 or 5 mm line and were available with either two or four stroke engines..
The video (vhs cassette) that came with each machine showed them being used in much the same situation as we would use a Hayterette rather than a strimmer replacement.
They are still around and a friend tells me that two are used to maintain his local churchyard.
Around about 2010 Hayter sold a similar Murray branded version.
 
 
 
 

Page 13 of today's Daily Telegraph shows a ride-on triple. Looks a bit like an old Horwool or Nickerson Turf Master or even  a Jacobsen. A two word name on the bonnet that appears not to match neither Wheel Horse nor John Deere.
Place your bets!
 
https://www.dropbox.com/s/2q73dve6ogy79ty/DT pic0001.pdf?dl=0

I take the point about the pin coupling but it could have started life with a cheepo pressed steel ball socket. Unless it was just a one off built with "what we had around us", I cant see that a commercial manufacturer would spend on Indespension units for a low speed off road application. Garden trailers of the day, such as Saxon, made do with a 1" diameter bar for an axle and wheels with either plain or roller bearings.  

Will be both an interesting project and a useful trailer but with those Indispension units,  was it originally built as  a " garden trailer"?  Possibly a basic lightweight car trailer ?

Brave man!
By the look of it you have a Peerless VST205 transaxle. Countax were the first OEM to fit that box in the UK and it was an unmitigated disaster. Your Hayter is possibly from a batch of machines made for them by Countax - standard K or C series machines painted green and usually with Kohler engines.
The attached service bulletin makes the point that dealers were advised not to undertake repairs but to swap out the complete box. The lack of quality control was such that some boxes never gave problems while others failed within days of being put to use . Sometimes a replacement box was so rough that it never left the workshop. Our record was two consecutive dud warranty replacements before finding one that we would let go back to the customer.
The claim that they didn't need a tow/dump valve was a bit wide of the mark and it wasn't long before they were produced with a mechanical disconnect kit to make it possible to move the machines without engine power.
Needless to say, Countax only used that box for one season.
This may be of interest
https://www.dropbox.com/s/6b6x1ihhrcns33q/Peerless VST2050001.pdf?dl=0
I think that I have some info somewhere - I'll have a look.
Edit Here's some more
https://www.dropbox.com/s/6pt795l6vksh1c7/PEERLESS 205vst Update0001.pdf?dl=0

The best solutions were on the old US Perr Forum which is no longer accessible but one of the former contributors has posted the gist of the solution on the current PPETEN forum - worth a read. If you are going the Pusa Jet repair kit route which will provide the necessary teflon washer the kits are still available under BS pt no 391413 or 4184 for the ten pack.
 
This may be of interest
http://outdoorpowerinfo.com/repairs/briggs_medium_two_piece.asp

Yes, they do split and an additional test is to hold it under just boiled water and watch for bubbles as the air  inside expands. Post a pic of the carb' , if it's a two piece flow jet they are known for flooding due to internal leaking where the angled emulsion tube screws in but doesn't seat properly. If you google Briggs Flow jet there is plenty of info.

Yes, they do split and an additional test is to hold it under just boiled water and watch for bubbles as the air  inside expands. Post a pic of the carb' , if it's a two piece flow jet they are known for flooding due to internal leaking where the angled emulsion tube screws in but doesn't seat properly. If you google Briggs Flow jet there is plenty of info.

Legged pullers that hook under the flywheel are a strict NO, NO. You can end up with a split flywheel or worse still a cracked one that flies apart later. 
Make what you need when you need it  - will always come in useful!

Legged pullers that hook under the flywheel are a strict NO, NO. You can end up with a split flywheel or worse still a cracked one that flies apart later. 
Make what you need when you need it  - will always come in useful!

Line 'em up on a well ventilated shelf, and hook each to a "maintenance" charger once a month. keeping them charged protects against sulphation and frost.
I have my stand-by generator permanently hooked to a C-tek charger .

Line 'em up on a well ventilated shelf, and hook each to a "maintenance" charger once a month. keeping them charged protects against sulphation and frost.
I have my stand-by generator permanently hooked to a C-tek charger .