Rice -Maize -Fallow is the potential cropping system, so is there any program to include legumes like cowpea, frenchbean etc. in the intercropping or relay yardlongbean with the maize stalk?
Maize cowpea intercropping can work well. If you can have a two row planter use row seed source in the cowpea and the other row in maize. Rather than use fallow try using a soybean crop previous to rice follow the maize cowpea intercrop. If you are rearing animals the corral manure can be composted with straw and applied on the maize interrow and on the rice crop. Using such a system will be able to eliminate the need for synthetic nitrogen application and improve soil organic matter.
I do not recommend dry bean or French vegetable beans because they are poor nitrogen fixers. Mung cowpea and yardlong are more aggressive fixers and better climbers. Use a hybrid maize well adapted to your conditions. The vignas have an advantage of fixing nitrogen with the bacteria common to your area but you could inoculate the seed to ensure best results. Vignas are also better than dry beans dealing with any weed issues.
The cropping pattern that you are asking about is common all over India. After the harvest of paddy, a crop of maize with pulse or only legume crop is taken, with the residual moisture present in the ground.