Sounds like you've ruled out the fungal possibilities like Sclerotinia and Fusarium spp. Blackleg bacterial disease seems unlikely in this case. Possibly Ralstonia solanacearum (closely related to Pseudomonas) - this is well known on potato, the tomato pith necrosis type organisms causing significant issues in potato might be a finding. Potato yellow dwarf virus is another possibility to check out,

Tim,

now, I have 3 facts:

1) this is a seed tubers infection (half of the field planted with other seeds stays healthy);

2) PCR indicated Dickeya and Pectobacterium, absence of  Ralstonia and Clavibacter sepedonicus, but nobody looked for Pseudomonas

3) Pseudomonads (P. marginalis, P. viridiflava, and P. corrugata-looking bacteria) were isolated almost pure from pith and necrotic lesions on the stems.

No fungi growth was found on upper parts of stems with lesions, I assume that white mycelium on some diseased stems was caused by saprophytic fungi from soil. Temperature was rather mild at first half of summer, and turned to hot (above 35oC) just before the symptoms occurrence.  I inoculated some potato plants in glasshouse, but now we have cold nights (around +3-5oC outside, and +15oC inside) and I am not sure  that it will be suitable for the disease development.

PS: Seeds were tested for Potato yellow dwarf virus, result was negative 

Excellent, narrowed down the options significantly and waiting for Koch's postulates if the weather allows it now. May have to provide more heated conditions. Are there symptoms on the tubers? Maybe you can do more controlled and quicker Kochs-postulate-like work on potato tuber slices though it won't be the same definite outcome as seeing stem necrosis.

P. corrugata has been reported in potato stems previously (e.g. Garbeva et al. 2001, link attached) but I'm not sure it (or P.marginalis, P. viridiflava) has been identified as a significant potato pathogen by anyone previously.

http://link.springer.com/article/10.1007/s002480000096#page-1

Tim, thanks! We had similar result by pure culture isolation and sequencing 16S rRNA (Pseudomonas marginalis & syringae groups, as well as other species listed by Garbeva et al.) for bacteria from diseased tubers at 2001-2004, but have never published it.

I have inoculated young tomatoes and hope they will get symptoms earlier (I used rooted secondary shoots of the one cultivar that is listed as pith necrosis- susceptible).

One type of the isolated bacteria gave wilt in 12 h after inoculation and maceration of low 1/3 of stem with black lesions at middle-upper part of stem in 36h. This is a picture made in 48h after inoculation of  tomato cuttings placed into bacterial suspension 10E8 CFU/ml  and kept at 15-20oC.  Bacteria colonies on King's B Agar looked small, transparent, no slime, non-pigmented, not-fluorescent, Gram(-), slow-growing at 24oC.