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For zero-curvature representations (ZCRs) At-Bx-AB+BA=0 of evolution equations ut=f(x, u, ux, ..., ux...x), we develop a description which is invariant under gauge transformations A'=SAS-1-SxS-1 and B'=SBS-1-StS-1, where A, B and S are matrix functions of x, u, ux, uxx, ... . We prove that every fixed matrix A of any dimension and order (in ux...x) determines a continual class of evolution equations which admit ZCRs with this A. Then we quote examples illustrating how a dependence of A on an essential parameter restricts classes of represented equations. One of our examples shows that some non-integrable systems can admit parametric Lax pairs and infinitely many non-trivial conservation laws.

Applying the first Noether theorem to the Liouville equation uxy=exp u, we find all (namely, a continuum of) non-trivial conservation laws of this equation. Then we find five new zero-curvature representations of the Liouville equation (by 2*2 traceless matrices) which contain, respectively, 1, 1, 2, 2 and 3 essential parameters. Finally, we show that all known zero-curvature representations of the Liouville equation are equivalent (in a definite sense) to matrices of conservation laws.

The singularity analysis of the system of nonlinear equations iat=axx+aa2a*-ip, px+i beta p+ar=0, rx= 1/2 (ap*+a*p) (where * denotes the complex conjugation, functions a and p are complex, function r and constants alpha and beta are real) indicates that the system has the Painleve property at a= 1/2 only. This analytic exclusiveness of the case a= 1/2 agrees with results by Doktorov and Vlasov (1983) who selected the same case by a modification of the Wahlquist-Estabrook method and found a corresponding Lax pair. In the integrable case, the method of truncating Weiss-Tabor-Carnevale expansions determines a Backlund auto-transformation which, unfortunately, violates the condition of complex conjugateness between a and a*. Another Backlund auto-transformation, compatible with this condition, is found by a technique of Miura transformations.

The general Miura transformation (t,x,u(t,x)) to (s,y,v(s,y)): v=a(t,x,u,. . ., delta ru/ delta xr), y=b(t,x,u,. . ., delta ru/ delta xr), s=c(t,x,u,. . ., delta ru/ delta xr) is considered which connects two evolution equations ut=f(t,x,u,. . ., delta nu/ delta xn) and vs=g(t,x,u,. . ., delta mu/ delta xm). The conditions c=c(t) and m=n are proven to be necessary. It is shown that every Miura transformation, admitted by a constant separant equation ut=f, consists of the following three transformations: (i) (t,x,u) to (t,x,w), where w=a(t,x,u,. . .,ux. . .x); (ii) (t,x,w) to (t,y,v), where y=x and v=w, or y=w and v=wx, or y=wx and v=wxx; (iii) a transformation of time s=c(t) and a contact transformation of (y,v). As an example, the Korteweg-de Vries equation is transformed to three new nonlinear equations, of which two have neither nontrivial algebra of generalized symmetries nor infinite set of conserved densities.

Noteworthy analytic properties of the Dym-Kruskal equation, such as the weak Painleve property and the Pogrebkov property (a second-order pole at infinity), are proven to be exactly the image of the usual Painleve property under the Ibragimov transformation. The transformed Painleve analysis is carried out for two infinite sets of evolution equations similar to the Dym-Kruskal equation. Only four of the equations pass the test, being known and integrable.

All Klein-Gordon, omega xy=f ( omega ), and evolution, omega t=f ( omega , omega x,. . ., omega x.7T. .$x), equations admitting autotransformations phi =a( omega ) are found (a( omega ) is such a function of omega and its finite-order derivatives that any solution omega is mapped into a solution phi of the same equation). Besides the linear ones, they include only the Liouville equation and the Burgers equation hierarchy. Their special Backlund autotransformations phi =a( omega ) are also found.

We perform the singularity analysis of four new generalized bilinear equations by Hu (1993), of which two possess two-soliton solutions and the other two possess even three-soliton solutions. All four equations fail to pass the Painleve test for integrability, exhibiting non-dominant movable logarithmic singularities at highest resonances of generic branches.

New Fujimoto-Watanabe non-constant separant evolution equations admitting generalized symmetries are shown to be connected with the Korteweg-de Vries, Sawada-Kotera, Kaup and linear equations via chains of differential substitutions. A conjecture is proposed which explains why the Ibragimov transformation to constant separant equations is a universal link in those chains.

The Thomas equation has the Lie-Backlund algebra recurrence operator of the first order, the infinite set of the many-parameter Backlund autotransformations and no reduction to the Painleve transcendents. The Burgers equation has the same properties which are probably the general indicators of the hidden linearity.

The applicability of the Painleve test of the complete integrability of the one-component nonlinear Klein-Fock-Gordon equations in an arbitrary Riemannian space, in the formulation of Weiss et al. (J. Math. Phys., vol.25, p.13, 1984) is discussed. Three infinite series of these equations are found in the flat two-dimensional space which possess the Painleve property and include, as a special case, the Liouville, sine-Gordon, and Dodd-Bullough equations. It is pointed out that the approach of Weiss et al. to select integrable nonlinear equations is not sufficiently reliable and needs some strengthening.