A general theory for the existence of solitary structure at M = Mc has been discussed, where Mc is the lower bound of the Mach number M, i.e., solitary structures start to exist for M > Mc. Three important results have been proved to confirm the existence of solitary structure at M = Mc. If V(φ)(≡ V(M,φ)) denotes the Sagdeev potential with φ being the perturbed field or perturbed dependent variable associated with a specific problem, V(M, φ) is well defined as a real number for all M ∈ ℳ and φ ∈ Φ0, and V(M, 0) = V′(M, 0) = V″(Mc, 0) = 0, V‴(Mc, 0) < 0 (V‴(Mc, 0) > 0), ∂ V/∂ M < 0 for all M (∈ ℳ) > 0 and φ (∈ Φ0) > 0 (φ (∈ Φ0) < 0), where ‘′ ≡ ∂/∂φ’, the main analytical results for the existence of solitary wave or double layer solution of the energy integral at M = Mc are as follows. Result 1: If there exists at least one value M0 of M such that the system supports positive (negative) potential solitary waves for all Mc < M < M0, then there exists either a positive (negative) potential solitary wave or a positive (negative) potential double layer at M = Mc. Result 2: If the system supports only negative (positive) potential solitary waves for M > Mc, then there does not exist positive (negative) potential solitary wave at M = Mc. Result 3: It is not possible to have coexistence of both positive and negative potential solitary structures at M = Mc. Apart from the conditions of Result 1, the double layer solution at M = Mc is possible only when there exists a double layer solution in any right neighborhood of Mc. Finally, these analytical results have been applied to a specific problem on dust acoustic (DA) waves in non-thermal plasma in search of new results.