Alright, I suppose that Sol could reopen the node (all evidence is that Earth was not destroyed shortly after the Lucifer was destroyed), they have the resources to possibly do it. However, I doubt they would. Remember, though Sol was the center of the GTA, all that knowledge was likely spread throughout the Alliance, and the Vasudans had their own sources. So, if they couldn't come up with a way to stabalize nodes in some 30 years, and they were trying, what are the odds that Sol could pull it off on its own? Not very good, I believe.
In any case, I doubt Sol would even try. venom says that cleaning up the Shivans was easy, but did you really remember the end of the game? The Shivans were everywhere having blockaded everything past Beta Aquilea. The GTA was on the verge of exitinction by this point, most of their best ships and pilots were dead, millions of refugees from Vasudan and Terran worlds crowded the systems still in Allied hands, and there appeared to be little chance of forcing them back.
It was only after the Lucifer was destroyed did the Shivans lose their coecion that made them the most deadly force in the galaxy. The people of Sol wouldn't know that. Communication with the rest of the GTA was completely cut off, and this fact probably was unknown to Sol. So in reality, I doubt Sol would even attempt such a task, as it would waste resources and possibly lead to their own demise. Remember, they likely don't know that the GTVA is alive and well, and it would be safer, for them, to stay cut off and prepare for the worst.