Based on a representation in terms of determinants of the order $2N$, we attempt to classify quasirational solutions of the one-dimensional focusing nonlinear Schrodinger equation and also formulate several conjectures about the structure of the solutions. These solutions can be written as a product of a t-dependent exponential times a quotient of two $N(N+ 1)$th degree polynomials in x and t depending on $2N-2$ parameters. It is remarkable that if all parameters are equal to zero in this representation, then we recover the $P_N$ breathers.