The natural parameterization of the exponential class of densities always exist? Def. Exponential Class of Densities
The density function $f\left ( \mathbf{x};\mathbf{\Theta} \right )$ is a member of the exponential class of density functions iff
$$f\left ( \mathbf{x};\mathbf{\Theta} \right )=\begin{cases}
 \exp\left [ \sum_{i=1}^{k}c_{i}(\mathbf{\Theta})g_{i}(\mathbf{x})+d(\mathbf{\Theta})+z(\mathbf{x})) \right ]& \text{for}\quad \mathbf{x}\in A,\\
 0& \text{ otherwise } , 
\end{cases}
$$
where $\mathbf{x}=(x_1,...,x_n)^{'},\mathbf{\Theta}=(\theta_1,...,\theta_k)^{'};c_{i}(\mathbf{\Theta}),i=1,...,k,$ and $d(\mathbf{\Theta})$ are real-valued functions of $\mathbf{\Theta}$ that do not depend on $\mathbf{x};g_i(\mathbf{x}),i=1,...,k,$and $z(\mathbf{x})$ are real-valued functions of $\mathbf{x}$ that do not depend on  $\mathbf{\Theta};$ and $A\subset\mathbf{R}^{n}$ is a set of $n$-tuples contained in $n$-dimensional real space whose definition dose not depend on the parameter vector $\mathbf{\Theta}.$

Considering an alternative parameterization of the exponential class of densities
$$f_{*}\left ( \mathbf{x};\mathbf{c} \right )=\exp\left [ \sum_{i=1}^{k}c_{i}g_{i}(\mathbf{x})+d_{*}(\mathbf{c})+z(\mathbf{x})) \right ]I_{A}(\mathbf{x}),\mathbf{c}=(c_1,...,c_k)\in \Omega_{\mathbf{c}},$$  with $$d_{*}(\mathbf{c})=\log\left [ \int_{\mathbf{R}^n}\exp\left\{  \sum_{i=1}^{k}c_{i}g_{i}(\mathbf{x})+z(\mathbf{x})) \right\}I_{A}(\mathbf{x})\mathrm{d}{\mathbf{x}}\right ]^{-1}$$
(use summation in the discrete case). This parameterization is
referred to as the natural parameterization of the exponential class of densities. Note that the definition of $d_{*}(\mathbf{c})$ is a direct result of
the fact that the density must integrate (or sum) to $1$.

The natural parameterization of an exponential class of densities maybe not unique.
I have a very elementary problem:Is there such an exponential class of densities its natural parameterization does not exist?

$[\rm I]$ defines the exponential family with the densities of the form $$ \exp\{A(x)B(\theta)+C(x)+D(\theta)\};\tag 1\label 1$$ $\eqref 1$ is natural exponential family if $A(\cdot) $ is linear.
Consider $X$ following lognormal distribution. Then one can show $A(x) =\ln x$ and thus a lognormal family is not natural exponential family.

Reference:
$\rm [I]$ Unifying the Named Natural Exponential Families and Their Relatives, Carl N. Morris, Kari F. Lock, The American Statistician, August $2009,$ DOI: 10.1198/tast.2009.08145.


Canonical forms are the building (cf.$\rm [I]$) blocks of exponential family of distributions. Formally, with a $\sigma$-finite measure $\mu$ on $\mathbb R^n,$ a non-negative real valued function $h:\mathbb R^n\to \mathbb R, $ a set of $\boldsymbol{\mathfrak B}(\mathbb R^n) /\boldsymbol{\mathfrak B}(\mathbb R) $ measurable functions $\langle T_i\rangle_{1\leq i\leq s},$ for $\boldsymbol\eta\in\mathbb R^s,$ let $$ \Xi:=\{\boldsymbol\eta:A(\boldsymbol\eta)<\infty\}$$ where $$ A(\boldsymbol\eta) =\ln\int \exp\left[\boldsymbol\eta^\top\mathbf T\right]h(\mathbf x) ~\mathrm d\mu.$$ Then the density $$p(\mathbf x;\boldsymbol\eta)=\exp\left[\boldsymbol\eta^\top\mathbf T-A(\boldsymbol\eta)\right]h(\mathbf x) \tag 1$$ is an $s$-parameter exponential family in canonical form (cf. $\rm [II]$).
$\Xi$ is the maximal parameter (cf.$\rm [III]$) space.
One can make other parameterizations of the density by taking a mapping of the form $\boldsymbol\theta\mapsto \boldsymbol \eta(\boldsymbol\theta) $ from some space $\Omega$ to $\Xi$ to get $$p(\mathbf x;\boldsymbol\theta)=\exp\left[\boldsymbol\eta(\boldsymbol\theta)^\top\mathbf T-B(\boldsymbol\theta)\right]h(\mathbf x) \tag 2$$ where $B(\boldsymbol\theta) := A(\boldsymbol\eta(\boldsymbol\theta)). $ Family $\{p(~;\boldsymbol\theta):\boldsymbol\theta\in\Omega\}$ is an $s$-paramter-exponential family.
$\rm [IV]$ formalizes the process of building up an exponential family (for simplicity, we will be dealing with families with parameter one):
$\bullet$ Start with a density $h$ w.r.t.  the $\sigma$-finite measure $\mu.$
$\bullet$ By exponential tilting, construct a family of densities proportional to $\exp(\eta x)h(x);$ consider  $M(\eta) = \int_\mathcal X e^{\eta x}h(x)~\mathrm d\mu;$ if $M(\eta)$ is finite for $\theta$ in a nhood of $0,$  it is the mgf of $h(x).$ Take $\Xi:= \{\theta\in \mathbb R: M(\theta)<\infty\}.$ If $\Xi$ is not degenerate, construct the family $\mathcal F_{ne}^1$ by $\eta \mapsto \exp\{\eta x- A(\eta)\}h(x),$ where $A(\eta)$ is the cumulant generating function of the distribution with pdf $h(x).$
$\bullet$ Let $T(x)$ be a statistic with cgf $A_T(\eta), ~\eta\in \Xi_T\subset\Xi.$ Then $\exp\{\eta T(x)-A_T(\eta)\}h(x)$ is a density for $\eta\in \Xi_T.$
$\bullet$ Consider a mapping $\Phi\to \Xi_T:\phi\mapsto\eta(\phi).$ Then the class of densities $$\exp\{\eta(\phi)T(x)- A_T(\eta(\phi))\}h(x) \equiv \exp\{\eta(\phi)T(x)- B(\phi)\}h(x)$$ is (full if $\eta(\Phi) = \Xi$) exponential family, denoted by $\mathcal F_e^1.$
$\bullet$ For each $\mathcal F_{ne}^1,$ there is an associated $\mathcal F_e^1$ induced by $T(\cdot)$ and parameterized by $\eta = \eta(\phi),$ that is $\exp\{\eta t- B(\phi(\eta))\}h_T(t)$ where $\eta(\phi)$ is assumed to be one-to-one with inverse $\phi(\theta).$
In a nutshell, as Wikipedia states

By defining a transformed parameter $\eta = \eta(\theta)$, it is always possible to convert an exponential family to canonical form.


References:
$\rm [I]$ Mathematical Statistics: Basic Ideas and Selected Topics- Vol. $\mathsf I, $ Peter J. Bickel, Kjell A. Doksum, Taylor & Francis, $2015, $ sec. $1.6.3, $ p. $56.$
$\rm [II]$ Theoretical Statistics: Topics for a Core Course, Robert W. Keener, Springer Science$+$Business, $2010, $ sec. $2.1, $ pp. $25-26.$
$\rm [III]$ Theory of Statistical Inference, Anthony Almudevar, Taylor & Francis, $2022, $ sec. $3.6, $ p. $73.$
$\rm [IV]$ Principles of Statistical Inference: From a Neo-Fisherian Perspective, Luigi Pace, Alessandria Salvan, World Scientific, $1997,$ sec. $5.2,$ pp. $172, 175-176.$
