  We begin by introducing a new procedure for construction of the exact
solutions to Cauchy problem of the real-valued (hyperbolic) Novikov-Veselov
equation which is based on the Moutard symmetry. The procedure shown therein
utilizes the well-known Airy function $\Ai(\xi)$ which in turn serves as a
solution to the ordinary differential equation $\frac{d^2 z}{d \xi^2} = \xi z$.
In the second part of the article we show that the aforementioned procedure can
also work for the $n$-th order generalizations of the Novikov-Veselov equation,
provided that one replaces the Airy function with the appropriate solution of
the ordinary differential equation $\frac{d^{n-1} z}{d \xi^{n-1}} = \xi z$.
