  A complete set of solutions |z,u,v>_{sa} of the eigenvalue equation
(ua^2+va^{dagger 2})|z,u,v> = z|z,u,v> ([a,a^{dagger}]=1) are constructed and
discussed. These and only these states minimize the Schr\"{o}dinger uncertainty
inequality for the squared amplitude (s.a.) quadratures. Some general
properties of Schr\"{o}dinger intelligent states (SIS) |z,u,v> for any two
observables X, Y are discussed, the sets of even and odd s.a. SIS |z,u,v;+,->
being studied in greater detail. The set of s.a. SIS contain all even and odd
coherent states (CS) of Dodonov, Malkin and Man'ko, the Perelomov SU(1,1) CS
and the squeezed Hermite polynomial states of Bergou, Hillery and Yu. The even
and odd SIS can exhibit very strong both linear and quadratic squeezing (even
simultaneously) and super- and subpoissonian statistics as well. A simple
sufficient condition for superpoissonian statistics is obtained and the
diagonalization of the amplitude and s.a. uncertainty matrices in any pure or
mixed state by linear canonical transformations is proven.
