Summary: We obtain some explicit expressions about rooted petal bundles on the orientable surfaces (genus 2 and 3) and the nonorientable surfaces (genus 5) with the size as a parameter. Meanwhile, we consider a kind of maps: nearly 2-regular maps, topologically equivalent to petal bundles. Using combinatorial method, we derive an explicit expression of rooted nearly 2-regular maps on the sphere with three parameters and the number of 2-regular maps on the surfaces with given genus.