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In the type~1 case, $f(x,y,\\cdot)$ is a general strictly increasing function; this is a mildly degenerate case. In the type~2 case, $f(x,y,\\cdot)$ has the form $h(x,y)F(u)+S(x,y)$, where $F(u)$ is just a nondecreasing function; this is a strongly degenerate case. 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