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Unfortunately to deal with a non-performing partner you have to be somewhat flexible in your approach.
There is no mechanism to simply push out a partner unless you actually have a partnership agreement (partnership), by-laws (corporation), or operating agreement (LLC) that allows an owner to be removed at the will of the other owners - in which case these documents will give you that procedure and you must follow it exactly.
But in most cases, the two options that you have open to you are: (1) buy the partner out (so you are going to have to come up with a price that everyone agrees on); or (2) sell the company, liquidate the assets, pay off the debts, and then divide the equity/remainder (then the two remaining partners can start over).
You can force the dissolution of a company by going to court if none of the partners will agree on either of these two remedies.
Short of filing a lawsuit, you can try to mediate the dispute with them - contact your local bar association and request referrals to mediators, a third party neutral can often help you reach a mutually agreeable resolution. Use the bar association's referrals to contact a mediator or two, the mediator will then contact the other party to set up a mediation session, and you can go from there - hopefully resulting in a formal or written settlement agreement, and save yourself the time and expense of litigation.
Mediation is usually a good tool for resolving these kinds of disputes, as it allows you to be flexible in resolving the underlying problems as well as addressing ways to buy out or change the ownership model so that the business is not immediately crippled by a cash flow problem.