In this paper we investigate the numerous possible parameter choices for linear lattice Boltzmann schemes according to the definition of the order of isotropy given in Augier et al. (in press) [4]. This property-written in a general framework that includes all of the DdQq schemes-can be understood through a group operation. The resulting relationships between the parameters of the scheme (defining the equilibrium states and relaxation times) yield a rigorous methodology that should be followed if one is to ensure isotropy at a given order. For acoustic applications in two spacial dimensions (namely D2Q9 and D2Q13 schemes) this methodology is used to propose a full description of the sets of parameters that are isotropic at order m (m1,2,3,4,5 for D2Q 9 and m1,2 for D2Q13). We include numerical illustrations for the D2Q9 scheme obtained with the code LBMpy developed in the laboratory of Mathematics of the University Paris-Sud.