Alexian Brothers Health System and Adventist Midwest Health on Tuesday said they are pursuing an affiliation that would create the second-largest hospital system in the state. The systems have signed a letter of intent to form a joint operating company that would oversee nine Chicagoland hospitals, five owned by Alexian and four by Adventist. The tie-up stops short of an outright merger; both local systems will maintain separate assets and balance sheets, in part because each is owned by national hospital chains with different religious affiliations.