Suppose that $(x, y)$ is a solution to $x^2 - 15y^2 = 1$. Then $(a, b, c) = (31, 5x, 15y)$ satisfies $2a^2 + 3b^2 - 5c^2 = 1997$. But since $x^2 - 15y^2 = 1$ is a Pell equation, there are infinitely many triples $(a, b, c)$.