As you have probably found, Russian Vine (Fallopia baldschuanica) can be very invasive - its other common name is Mile-a-Minute plant! Firstly, cut the plant back hard to just a few feet above ground level. Removing all of that top growth will weaken it and make it more manageable for you. If you are looking for an organic solution then your best bet is to dig out the roots. You will need to keep an eye out for any regrowth which should be removed in the same way. Otherwise, use a systemic glyphosate based herbicide such as Round Up which can be painted on to the leaves. You may need a couple of applications though.
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