In this paper we shall study relations between three generalisations of CMC surfaces-- Bonnet surfaces, $H$-surfaces and HIMC surfaces. We introduce the notion of HIMC surfaces in space forms and show that HIMC surfaces have similar properties to CMC surfaces. Furthermore we shall introduce a special one-parameter family of framings for $H$- surfaces in the space forms. Through these framings we reveal relationship between the associated family of $H$-surfaces and the extended solutions for harmonic maps into SO(3). .