This paper provides a new analytic test to check strong accessibility of nonlinear control systems. This test can be applied to nonlinear systems described by meromorphic vector fields (whose components are fractions of analytic functions). The test consists in checking the rank of a certain finite matrix (which extends to a nonlinear case the classical "controllability matrix" of linear systems theory) and, unlike the classical accessibility test, always terminates giving an exhaustive answer (yes/no) within a fixed number of steps. Further, a generic feature of accessibility is proven: the accessibility from a point $p$ of the system analyticity domain implies the accessibility from almost every point of the domain (generic strong accessibility). The paper provides two simple examples, which illustrate the proposed analytic test and its advantages with respect to the existing results.