"You could have a person who owns a bobcat with the auger bit drill holes (from the outside) diagonally under the wall down to bedrock or supportive clay and then pump them full of concrete. Then, locate the poured beams inside the basement and place wooden beams vertically over them, using jacks to force those beams against the floor joists above. This will lift the sagging structure, and allow rebuilding of the wall in question. Depending on your local codes the vertical wooden beams could remain in place for support."
Diagonally beside the basement or towards the basement.....I may need a stickman kind of drawing on this.
To clear it up a bit, the heaving is between mine and my neighbour's house (all of three feet between) on the north wall. It's heaving out not in only on that wall. The west wall is the one sinking. It happened fast too. Kinda scary!