Conservation laws for one-dimensional isentropic gas flows

The governing equations of one-dimensional isentropic gas flow are expressed in terms of a pair of exterior differential forms. By employing Cartan's theory and the classical Frobenius theorem linear partial differential equations are obtained to determine conservation laws which are conjectured to be the key to detect completely integrable systems. By using a similarity technique explicit expressions are provided for polynomial type of conservation laws in terms of Gegenbauer and Chebyshev polynomials.