[pescetom]

Our 2/1 system has a very natural base and defines the auction
1♥-1♠
2♥-3♠
as Responder showing 6+ spades, invitational, NF, does not deny or affirm hearts fit.

I've always been uncomfortable with that and feel it should either be game forcing or deny hearts fit (or affirm it, but that is not consistent with our system or style).
How do others play it, or wish to play it?
Is there some convention over 2♥ that might lubricate the situation?

[DavidKok]

This is one of my favourite bidding questions, because there's such elegant natural solutions. I'll give two different solutions that I suspect might not require much or even any system modification, and then go into a bit more detail on my personal preferences and some considerations here.

Firstly, I feel that 3♠ should deny a heart fit. You don't need to find a double major suit fit if you've already found the hearts, you can't play in both strains at the same time anyway. If we have a known heart fit on this start I would take the money and raise partner.
Secondly, when partner opens at the 1-level and we have a 6(+)cM in a higher ranking major (so not us having hearts over a 1♠ opening, but all other situations apply), we have two ways to get to 2M: directly and via 1M. We also have two hand types with a long major that we wish to describe cheaply: weak and invitational (because game forcing can bid 4SGF/3SGF/XYNT/other to show the hand eventually). I really like picking one of the two ranges and putting in the direct bid, here 1♥-2♠, and taking the other range and putting it in the delayed bid, here 1♥-1♠; 2♥-2♠. Either way, we get to show the invitational hand with spades and no fit at the 2♠ level. No need to go to 3♠, let alone with a jump. No need for conventions either, these bids are all natural and NF. This way we can reserve the 3♠ rebid for something else - I like game forcing, slam-going, sets trumps, demands a control bid.
Either of these two agreements would resolve most of your issue and not require new conventions, I think.

Now a little bit more about this treatment:

• Using the 1m-2M bids and 1X-2Y jump shifts in general, there is a general principle that I think is poorly understood. If opener was about to rebid NT, be it 1NT or 2NT, we are usually in a comfortable position having responded 1M. Instead it is the suit rebids that make things awkward - taking away more bidding space, while being less descriptive with regards to strength and degree of fit (i.e. length in unbid suits). Therefore I feel strongly that the jump responses to 1-level openings should cater to the hands that are difficult to show on the 1X-1Y; 2Z auctions. The single-suited hands match this description well, and Reverse Flannery does to a degree too. Personally I really like having a way to invite in a major suit at the 2-level and getting the trump-setting jump rebids for free, so I'll stick with the single-suited meaning.
  
• If opener regularly ('always') raises on a 3-card suit on the auction 1X-1Y with an unbalanced hand, as is my preferred style, you have a negative inference on the example auction 1♥-1♠; 2♥ that partner has at most two hearts, to go with at most three clubs, three diamonds and at least six hearts. That's a pretty specific hand shape already - 2=6=3=3 minus a non-heart card. In view of that I think having to jump to 3♠ to invite in the hopes of finding extras and/or a doubleton opposite is not attractive at all. It's good to discuss with your partner under which conditions they would/wouldn't raise with 3-card support, and under which conditions they would/wouldn't bypass a 4cm.
  
• If we're using 1X-2Y (jump shift) as natural, I prefer for the direct jump to be weak (4-8) and the delayed bid to be invitational (9-11). The reasons for this are threefold:
  • If we have a weak hand the fourth seat is more likely to interfere, so we want to get our hand out of the way at once.
    
  • If opener has a strong hand, especially a strong misfit hand, the jump rebid auctions 1X-1Y; 3Z or the jump suit rebid auction 1X-1Y; 3X can be really awkward with a weak hand, since we'd really like to play in our own long suit to create more entries.
    
  • Invitational hands are often more flexible with regards to strain than weak hands, so by taking the slow action and getting more information we have a higher chance of finding a profitable alternative strain.
• It's already included above, but repeated for emphasis: the decision of how to play 1♥-2♠, and more generally 1X-2Y (jump shift), should not depend on your NT system or checkback variant. Those systems are all sufficiently airtight that you can show weak, invitational and strong at the 2-level or so. It's the unbalanced hands where you need to describe your single-suited hand quickly, or risk facing a tough decision next round.
  
• On 1♥-1♠ specifically, if your 1♠ response may still have a 3(+)c♥ support, you may well struggle to show this hand later. For that reason I prefer to eliminate hands with a fit as much as possible from changes of suit - support with support. In my system the exception is hands with slam interest, there I'm systemically forced into giving a delayed raise.

[mw64ahw]

Kaplan Inversion sorts this one out for me.

1♥ - 1♠ 0-4♠ weak, 5+♠ GI+, 3♥ lim
Now after 2♥
.. 2♠ 6+♠ GI+
.. 2N 5♠ GI
.. 3m 4m5♠ GF
.. 3♥ 2/3♥ min.
.. 3♠ self-sustaining ♠
.. 3N 5233 GF
.. 4♥ 2/3♥ max.

If you want to cater for 55 in the minors then
3♣ xs55
3♦ 5s(4x) GF, 3♥ asks