I have only heard the Ayon Triton once at a dealer so I can't comment there, but I do own an LSA Statement. The LSA is a hybrid amp, so you are essentially getting a tubed stage and a solid state amp. The LSA follows a design principle that says "one tube in the signal path per channel." i dont know much about this principle, but some people swear by it and ive found great success with it. While the LSA responds well to tube rolling in changing some of the midrange characteristics, it also has authoritative control of the music that you would expect out of the amp design.
I've gone through a few amps and none of them can match the LSA's ability to control bass while also producing clean and correct tone across all frequencies. lThe result is a linear and "true" tonality to my ear. If you listen to complex music like symphony, you are rewarded with hearing the whole performance (something my tube amps didn't do well), but it still swings on my acoustic jazz LPs. Now, this isn't to say that a great tube amp like the Triton can't give you many of these characteristics, but it isn't typical of my experience.
In the end, my opinion is that the LSA has managed to meld the best of two worlds (tone & sweetness, depth & control) with the drawbacks of neither (compressed sounding, edgy sounding). The Amp drives my ProAcs with ease (better than other amps I have had), and I have heard them drive far bigger at audio shows. I wholeheartedly recommend the LSA Statement to anyone looking for one of the best integrateds available. You won't regret it.
Good luck with your search!