Eventually I managed to make the antenna work on more than one band. I used a modeller (CocoaNEC) to approximately determine the impedance on each band of interest for my antenna configuration (inverted-V on a 10m long fishing rod). The impedance was about 2450 Ohm. Thus I rewinded the transformer to 1:49. Also I used a 1:1 balun (8 turns of RG58 on FT240-31 core) to eliminate any common mode current. Using a variable capacitor I've found a capacitance (138 pF in my case) that gives the best SWR plot:
No counterpoise was needneeded since it's role was played by the coax in the 1:1 balun. Then I replaced a variable capacitor with a constant NP0 capacitors. You can find a little more details here. The article is in Russian, but Google Translate should manage.
The antenna was tested on all band where it has SWR < 3: from 80m to 15m. QSOs were made on all these band. However, subjectively the overall performance of the antenna is not great comparing to the performance of a regular dipole. I wouldn't recommend trying to repetrepeat it, at least definitelyly not with cores I've used.