The word canonical means "by a general law, rule, principle or criterion". When the Hamiltonian operator is applied to the (average momentum) wave function it gives quantized values. In this sense the Hamilton equations gives the Schrodinger equation discreet values by a general law.
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They are called simultaneous equations.
A system of equations.
No.
A system of linear equations that has at least one solution is called consistent.
Statements in which the two sides are not equal are called inequalities.