The only international airport in the Solomon Islands is Henderson Field, served by the following airlines.
Air Pacific (http://www.airpacific.com/) once weekly service from Nadi
Air Niugini (http://www.airniugini.com.pg/) twice weekly to Port Moresby and twice weekly to Nadi
Solomon Airlines (http://www.flysolomons.com/) 4/5 times weekly service to Brisbane and Our Airline flying to Brisbane three times per week.
Pacific Blue (http://www.pacificblue.com.au) have flights to Brisbane every Tuesday and Thursday.