Help with an oil leak

Hi everyone
I purchased a T3 850 last year; haven’t ridden it very much yet so decided to take it for a little run.
Gave it a few revs on a short section of dual carriageway and noticed the smell of burning oil, which I’ve never come across before while riding.
When I got to my destination, about 8 miles in total, I had oil dripping off the left side of the engine. Clean fresh oil as I’d topped it up about a month previously, but cannot find a leak. Oil was on the left exhaust, foot peg, gear change pad, and on top of the air inlet to the left cylinder. Also noticed drops of oil on the interconnected fuel lines.
Rode it home gently and it didn’t drip once.
Dripped a few more drops over the space of one week in my garage but I think this was the last few drops finding their way down to the ground. Have cleaned around as much of the engine as I can easily get to, had the tank off to get a better look and I cannot see anywhere that oil is leaking from.
Any ideas?
Also found a rubber hose that appears to come out of the gear box, but goes nowhere. Some kind of breather? Advice on where it should go gratefully received! Thanks in advance.
Matt
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The gearbox breather should run up from the gearbox and just be an open pipe roughly level with the top frame tubes. Nothing more than a very fine mist comes from that.
I would check the oil feed pipe that sits between the barrels towards the front of the engine and goes up to feed the cylinder head rocker assembly. They have been known to fail and will spray oil as you describe.

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Absolutely. Pipe gets hot, pressure builds, split opens: oil exits. Pipe cools down, split seals, etc. Examine for condition and consider replacement anyway, that way it is ruled out, at least. Ditto rocker pipe breathers. If problems persists, perhaps share what type of oil collector/air box is fitted to your T3. These things get changed, not always effectively.

Thank you both for the information, I’ll have a closer look at the hoses; it would seem to make sense that it split under combined pressure and heat and then sealed up when the ride home was slow and gentle. Also, the position of those hoses you mentioned makes more sense as to how oil got so high up the engine!

Also look at rocker cover gaskets, and the blanking plugs inside.
if the plug leaks, oil runs down into the finning on the cylinder head.
Hot oil can travel a long way when the air blows it around so can be difficult to trace the source.

Check the oil pressure switch as well,mines just sprung a leak.

Thank you for the extra hints. I did suspect a blown gasket but having looked around the engine, I can see no obvious evidence of a leak from any of the mating surfaces and related gaskets, which is why it had me so stumped in the first instance. But, I will have another look for sure and see if the blanking plugs are an issue.

Thanks for the suggestion; I’ll have a look at that too. Hope your’s is fixed soon!

New oil pressure switch on it’s way

Well done. It’ll be the hoses (o; But at least you’ll have a spare pressure switch. Btw, don’t ever change that with the engine hot. Exhaust down pipes, that’s all I’m going to say.