Business attorneys do not do contingency cases. Contingency cases are the realm of such lawyers as ambulance chasers (aka personal injury), worker's comp and disability attorneys. A litigator that specializes in business matters would expect to be compensated for his work whether he wins, looses or settles a case for you. If it is a matter that does not involve litigation, then no attorney would provide his legal services to you on "contingency" as there is no contingency to base such representation upon.
The only possible scenario that I can see where an attorney would not get paid at the time of services for business law practice would be if that attorney is involved in a start-up company as an initial investor - where he provides his services for a stake in the company. But, unless you know an attorney that is willing to invest in your business, this is an unlikely scenario.
Depending upon the specifics of the litigation and your likelihood of winning, you may be able to get a general practitioner who will take on contingency cases to take on your case.
Common law is the body of law -- usually formed from court holdings and precedents -- that is enforced if there are not statutes (legislatively created laws) that would apply.
Basically -- common law started out as the body of what was commonly applied -- but has become the applicable court decisions that are binding on later and lower courts.