= Solution
Choose real constants $a_0<a_1<\cdots<a_m$ and define the smooth homogeneous quotient
$$
\boxed{f([x_0:\cdots:x_m])=\frac{\sum_{j=0}^ma_jx_j^2}{\sum_{j=0}^mx_j^2}.}
$$
It is unchanged by nonzero rescaling, so it is well defined on <Real projective space>. On the unit <sphere> its differential vanishes on the <tangent space> precisely when $Ax$ is proportional to $x$, where $A=\operatorname{diag}(a_0,\ldots,a_m)$. The entries are distinct, so the only critical projective lines are $p_i=[e_i]$. In the affine chart $x_i=1$, write $u_j=x_j/x_i$ for $j\ne i$. Then
$$
f=a_i+\frac{\sum_{j\ne i}(a_j-a_i)u_j^2}{1+\sum_{j\ne i}u_j^2}.
$$
Its <Hessian> at zero is diagonal with entries $2(a_j-a_i)$. Exactly $i$ are negative, and none vanish. Thus this is a <Morse function> with \b[one <critical point> $p_i$ of index $i$ for every $0\le i\le m$].
Choose the quotient of the round metric on the unit <sphere>. The downward <gradient flow> has <sphere> equation $\dot x_j=-2(a_j-f(x))x_j$ and projective solution
$$
[x_0(t):\cdots:x_m(t)]=[e^{-2a_0t}x_0(0):\cdots:e^{-2a_mt}x_m(0)].
$$
Normalization to the unit <sphere> supplies the common positive factor. As $t\to+\infty$, the smallest index with nonzero coordinate dominates; as $t\to-\infty$, the largest dominates. Consequently
$$
W^u(p_i)=\{x_i\ne0,\ x_j=0\ (j>i)\}\cong\mathbb R^i,\qquad W^s(p_i)=\{x_i\ne0,\ x_j=0\ (j<i)\}\cong\mathbb R^{m-i}.
$$
These are its <unstable manifolds> and <stable manifolds>. If $j\ge k$, the tangent directions in $W^u(p_j)$ allow coordinates $0,\ldots,j$, and those in $W^s(p_k)$ allow $k,\ldots,m$, modulo the common projective line. Together they span the full <tangent space> at every intersection. If $j<k$ the intersection is empty. Thus the flow is a <Morse-Smale gradient flow>, with no generic perturbation needed.
Orient each unstable <manifold> using its affine coordinates. The <Morse-Smale complex> has $C_i=\mathbb Z p_i$; its differential counts oriented unparametrized connecting trajectories of index difference one. A trajectory from $p_i$ to $p_{i-1}$ has only coordinates $x_{i-1},x_i$ nonzero, and there are exactly two, distinguished by the sign of $x_{i-1}/x_i$. Each sign branch is a single orbit modulo time translation.
To calculate their signs rather than merely count two, use the characteristic <closed disk> of the unstable cell: the closed upper hemisphere of $S^i$ with $x_i\ge0$, followed by projection onto $\mathbb{RP}^i$. Its interior is $W^u(p_i)$, and its boundary map is the antipodal double cover $S^{i-1}\to\mathbb{RP}^{i-1}$. After collapsing $\mathbb{RP}^{i-2}$, its degree on the $(i-1)$-cell is the sum of the degrees of the two sheets. Comparing the sheets by the <antipodal map> on $S^{i-1}$ gives relative orientation $(-1)^i$: it is induced by $-I$ on $\mathbb R^i$, whose determinant is $(-1)^i$. Thus the two trajectory signs add to $1+(-1)^i$, up to the choice of generator orientations. The <projective quadratic Morse complex> is therefore
$$
\boxed{C_i=\mathbb Z\ (0\le i\le m),\qquad\partial_i=\begin{cases}2,&i\text{ even},\\0,&i\text{ odd}.\end{cases}}
$$
In particular $\partial_1=0$, as required for the connected one-dimensional skeleton. The signed trajectory count agrees with this attaching-map incidence calculation, so it computes integral rather than only mod-two <homology>.
For $m\ge1$, taking kernels modulo images gives
$$
\boxed{H_k(\mathbb{RP}^m;\mathbb Z)=\begin{cases}\mathbb Z,&k=0,\\\mathbb Z/2,&0<k<m\text{ and }k\text{ odd},\\\mathbb Z,&k=m\text{ and }m\text{ odd},\\0,&\text{otherwise}.\end{cases}}
$$
For example, an odd interior-degree generator has zero outgoing differential but twice itself is the image from the next degree; an even positive-degree differential is injective. At the top there is no incoming differential, leaving a copy of $\mathbb Z$ exactly when $m$ is odd. For $m=0$ the space is one point and only $H_0=\mathbb Z$ remains. This recovers the <cellular homology of real projective space> by explicitly constructing its <Morse-Smale complex> and signs.
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